KUMMER SURFACES AND K3 SURFACES WITH - Mathématiques ...

KUMMER SURFACES AND K3 SURFACES WITH (Z/2Z)4 SYMPLECTIC ACTION ALICE GARBAGNATI AND ALESSANDRA SARTI Abstract. In the first part of this paper we give a survey of classical results on Kummer surfaces with Picard number 17 from the point of view of lattice theory. We prove ampleness properties for certain divisors on Kummer surfaces and we use them to describe projective models of Kummer surfaces of (1, d)polarized Abelian surfaces for d = 1, 2, 3. As a consequence we prove that in these cases the N´ eron–Severi group can be generated by lines. In the second part of the paper we use Kummer surfaces to obtain results on K3 surfaces with a symplectic action of the group (Z/2Z)4 . In particular we describe the possible N´ eron–Severi groups of the latter in the case that the Picard number is 16, which is the minimal possible. We describe also the N´ eron–Severi groups of the minimal resolution of the quotient surfaces which have 15 nodes. We extend certain classical results on Kummer surfaces to these families.

1. Introduction Kummer surfaces are particular K3 surfaces, obtained as minimal resolutions of the quotient of an Abelian surface by an involution. They are algebraic and form a 3-dimensional family of K3 surfaces. Kummer surfaces play a central role in the study of K3 surfaces, indeed certain results on K3 surfaces are easier to prove for Kummer surfaces (thanks to their relation with the Abelian surfaces), but can be extended to more general families of K3 surfaces: the most classical example of this is the Torelli theorem, which holds for every K3 surface. The aim of this paper is to describe some results on Kummer surfaces (some of them are classical) and to prove that these results extend to 4-dimensional families of K3 surfaces. Every Kummer surface has the following properties: it admits the group (Z/2Z)4 as group of automorphisms which preserves the period (these automorphisms will be called symplectic) and it is also the desingularization of the quotient of a K3 surface by the group (Z/2Z)4 which acts preserving the period. The families of K3 surfaces with one of these properties are 4-dimensional: we study these families using the results on Kummer surfaces and we prove that several properties of the Kummer surfaces hold more in general for at least one of these families.

2010 Mathematics Subject Classification. Primary 14J28; Secondary 14J10, 14J50. Key words and phrases. Kummer surfaces, K3 surfaces, automorphisms, Enriques involutions, even sets of nodes. The first author is partially supported by PRIN 2010–2011 “Geometria delle variet` a algebriche” and FIRB 2012 “Moduli Spaces and Their Applications”. Both authors are partially supported by GRIFGA: Groupement de Recherche Italo-Fran¸cais en G´ eom´ etrie Alg´ ebrique. 1

2

ALICE GARBAGNATI AND ALESSANDRA SARTI

The first part of the paper (Sections 2, 3, 4, 5) is devoted to Kummer surfaces. We first recall their construction and the definition of the Shioda–Inose structure which was introduced by Morrison in [Mo]. In particular we recall that every Kummer surface Km(A) is the quotient of both an Abelian surface and a K3 surface by an involution (cf. Proposition 2.16). Since we have these two descriptions of the same surface Km(A) we obtain also two different descriptions of the Néron– Severi group of Km(A) (see Proposition 2.6 and Theorem 2.18). In Section 3 we recall that every Kummer surface admits certain automorphisms, and in particular the group (Z/2Z)4 as group of symplectic automorphisms. In Proposition 3.3 we show that the minimal resolution of the quotient of a Kummer surface Km(A) by (Z/2Z)4 is again Km(A). This gives a third alternative description of a Kummer surface and shows that the family of Kummer surfaces is a subfamily both of the family of K3 surfaces X admitting (Z/2Z)4 as group of symplectic automorphisms and of the family of the K3 surfaces Y which are quotients of some K3 surfaces by the group (Z/2Z)4 . The main results on Kummer surfaces are contained in Section 4 and applied in Section 5: Nikulin, [Ni1], showed that a non empty set of disjoint smooth rational curves on a K3 surface can be the branch locus of a double cover only if it contains exactly 8 or 16 curves. In the first case the surface which we obtain by taking the double cover and contracting the (−1)-curves is again a K3 surface, in the second case the surface we obtain in the same way is an Abelian surface and the K3 surface is in fact its Kummer surface. In the sequel we call the sets of disjoint rational curves in the branch locus of a double cover even set. In [GSa1] we studied the Néron–Severi group, the ampleness properties of divisors and the associated projective models of K3 surfaces which admit an even set of 8 rational curves. Here we prove similar results for K3 surfaces admitting an even set of 16 rational curves, thus for the Kummer surfaces. In Section 4 we prove that certain divisors on Kummer surfaces are nef, or big and nef, or ample. In Section 5 we study some maps induced by the divisors considered before and we obtain projective models for the Kummer surfaces of the (1, d)-polarized Abelian surfaces for d = 1, 2, 3. As byproduct we show that these Kummer surfaces have at least one model such that their Néron–Severi group is generated by lines. Several models described are already well known, but here we suggest a systematical way to produce projective models of Kummer surfaces by using lattice theory. In the second part of the paper (Sections 6, 7, 8, 9, 10) we apply the previous results on Kummer surfaces to obtain general results on K3 surfaces X with symplectic action by (Z/2Z)4 and on the minimal resolutions Y of the quotients. In Theorem 7.1, Proposition 8.1 and Theorem 8.3 we describe explicitly N S(X) and N S(Y ) and thus we describe the families of the K3 surfaces X and Y proving that they are 4-dimensional and specialize to the family of the Kummer surfaces. In [Ke2] Keum proves that every Kummer surface admits an Enriques involution (i.e. a fixed points free involution). Here we prove that this property extends to every K3 surface X admitting a symplectic action of (Z/2Z)4 and with Picard number 16 (the minimal possible). This shows that the presence of a certain group of symplectic automorphisms on a K3 surface implies the presence of a non–symplectic involution as well.

KUMMER SURFACES, SYMPLECTIC ACTION

3

On the other hand certain results proved for Kummer surfaces hold also for the K3 surfaces Y . In Proposition 8.5 we prove that certain divisors on Y are ample (or nef and big) as we did in Section 4 for Kummer surfaces. The surface Y admits 15 nodes, by construction. We recall that every K3 surface with 16 nodes is in fact a Kummer surface; we prove that similarly every K3 surface which admits 15 nodes is the quotient of a K3 surface by a symplectic action of (Z/2Z)4 . This result is not trivial, indeed the following analogue is false: a K3 surface with 8 nodes is not necessarily the quotient of a K3 surface by a symplectic involution. Moreover we show in Theorem 8.3 that K3 surfaces with 15 nodes exist for polarizations of any degree (we give some examples in Section 10). This answers the question which is the maximal number of nodes a K3 surface with a given polarization can have (and it does not contain further singularities). If the polarization L with L2 = 2t has t even then the maximal number is 16 and it is attained precisely by Kummer surfaces, otherwise this maximum is 15. Finally, in Section 10 we give explicit examples of the surfaces X and Y and we describe their geometry. We point out that some results of the paper are (partially) contained in the first author’s PhD thesis, [G1]. Acknowledgments: We are indebted with Bert van Geemen for his support and invaluable help during the preparation of the paper. The Proposition 3.5 and the Theorems 8.3 and 8.6 are motivated by a question of Klaus Hulek and Ciro Ciliberto respectively. The study in Section 5 of K3 surfaces with Néron–Severi group generated by lines is motivated by several discussions with Masato Kuwata. We want to thank all of them for asking the questions and for their comments. 2. Generalities on Kummer surfaces 2.1. Kummer surfaces as quotients of Abelian surfaces. Kummer surfaces are K3 surfaces constructed as desingularization of the quotient of an Abelian surface A by an involution ι. Equivalently they are K3 surfaces admitting an even set of 16 disjoint rational curves. We recall briefly the construction: let A be an Abelian surface (here we consider only the case of algebraic Kummer surfaces), let ι be the involution ι : A −→ A, a 7→ −a. Let A/ι be the quotient surface. It has sixteen singular points of type A1 which are the image, under the quotient map, g be of the sixteen points of the set A[2] = {a ∈ A such that 2a = 0}. Let A/ι g is a K3 surface. the desingularization of A/ι. The smooth surface Km(A) := A/ι e Consider the surface A, obtained from A by blowing up the points in A[2]. The e whose fixed locus are the automorphism ι on A induces an automorphism e ι on A e ι is smooth. sixteen exceptional divisors of the blow up of A. Hence the quotient A/e e It is well known that A/e ι is isomorphic to Km(A) and that we have a commutative diagram: (1)

e A Km(A)

γ

/A πA

/ A/ι

e there are 16 exceptional curves of the blow up of the We observe that on A 16 points of A[2] ⊂ A. These curves are fixed by the involution e ι and hence are

4


mapped to 16 rational curves on Km(A). Each of these curves corresponds uniquely to a point of A[2]. Since A[2] ≃ (Z/2Z)4 , we denote these 16 rational curves on Km(A) by Ka1 ,a2 ,a3 ,a4 , where (a1 , a2 , a3 , a4 ) ∈ (Z/2Z)4 . Since the points in A[2] e are fixed by e are fixed by the involution ι, the exceptional curves on A ι and so the e cover A → Km(A). In curves Ka1 ,a2 ,a3 ,a4 are the branch locus of the 2 : 1 cyclicP particular the curves Ka1 ,a2 ,a3 ,a4 form an even set, i.e. 21 ( ai ∈Z/2Z Ka1 ,a2 ,a3 ,a4 ) ∈ N S(Km(A)). Definition 2.1. (cf. [Ni1]). The minimal primitive sublattice of H 2 (Km(A), Z) containing the 16 classes of the curves Ka1 ,a2 ,a3 ,a4 is called Kummer lattice and is denoted by K. In [Ni1] it is proved that a K3 surface X is a Kummer surface if and only if the the Kummer lattice is primitively embedded in N S(X). Proposition 2.2. [PS, Appendix to section 5, Lemma 4] The lattice K is a negative definite even lattice of rank sixteen. Its discriminant is 26 . Remark 2.3. Here we briefly recall the properties of K (these are well known and can be found e.g. in [PS], [BHPV], [Mo]): 1) Let W be a hyperplane in the affine 4-dimensional space (Z/2Z)4 , i.e. W P4 is defined by an equation of type i=1 αi ai = ǫ where αi , ǫ ∈ {0, 1}, and ai 6= 0 for at least one i ∈ {1, 2, 3, P4}. The hyperplane W consists of eight points. For every W , the class 12 p∈W Kp is in K and there are 30 classes of this kind.P 2) The class 12 p∈(Z/2Z)4 Kp is in K. 3) Let Wi = {(a1 , a2 , a3 , a4 ) ∈ A[2] such that ai = 0}, i = 1, 2, 3, 4. A setPof generators (over latticePis given by P the classes: P Z) of the 1Kummer P 1 1 1 1 p∈(Z/2Z)4 Kp ), 2 p∈W1 Kp , 2 p∈W2 Kp , 2 p∈W3 Kp , 2 p∈W4 Kp , 2( K0,0,0,0 , K1,0,0,0 , K0,1,0,0 , K0,0,1,0 , K0,0,0,1 , K0,0,1,1 , K0,1,0,1 , K1,0,0,1 , K0,1,1,0 , K1,0,1,0 , K1,1,0,0 . 4) The discriminant form of K is isometric to the discriminant form of U (2)⊕3 . In particular the discriminant group is (Z/2Z)6 , there are 35 non zero elements on which the discriminant form takes value 0 and 28 non zero elements on which the discriminant form takes value 1. 5) With respect to the group of isometries of K there are three orbits in the discriminant group: the orbit of zero, the orbit of the 35 non zero elements on which the discriminant form takes value 0 and the orbit of the 28 elements on which the discriminant form takes value 1. 6) Let V and V ′ be two 2-dimensional planes (they are the intersection of two hyperplanes in (Z/2Z)4 and thus isomorphic to (Z/2Z)2 ), such that V ∩ V ′ = {(0, 0,P0, 0)}. Denote by V ∗ V ′ := V ∪ V ′ − (V ∩ V ′ ), then the classes w4 := 12 p∈V Kp are 35 classes in K ∨ /K and the discriminant form P on them takes value 0; the classes w6 := 21 p∈V ∗V ′ Kp are in 28 classes in K ∨ /K and the discriminant form on them takes value 1,(see e.g. [G1, Proposition 2.1.13]). 7) Let Vi,j = {(0, 0, 0, 0), αi , αj , αi + αj } ⊂ (Z/2Z)4 , 1 ≤ i, j ≤ 4 where α1P = (1, 0, 0, 0), αP α3 = (0, 0, 1, 0),Pα4 = (0, 0, 0, 1). 2 = (0, 1, 0, 0), P P Then 1 1 1 1 1 K ), K ), K ), K ), ( ( ( ( ( p p p p p∈V1,3 p∈V1,4 p∈V2,3 p∈V2,4 Kp ), 2 Pp∈V1,2 2 2 2 2 1 p∈V3,4 Kp ) generate the discriminant group of the Kummer lattice. 2(


5

Here we want to relate the Néron–Severi group of the Abelian surface A with the Néron–Severi group of its Kummer surface Km(A). Recall that for an abelian variety A we have H 2 (A, Z) = U ⊕3 (see e.g. [Mo, Theorem-Definition 1.5]). Proposition 2.4. The isometry ι∗ induced by ι is the identity on H 2 (A, Z). Proof. The harmonic two forms on A are dxi ∧ dxj , i 6= j, i, j = 1, 2, 3, 4 where xi are the local coordinates of A viewed as the real four dimensional variety (R/Z)4 . ι∗

By the definition of ι we have: dxi ∧ dxj 7→ d(−xi ) ∧ d(−xj ) = dxi ∧ dxj . So ι induces the identity on H 2 (X, R) = H 2 (X, Z) ⊗ R and hence on H 2 (X, Z) since H 2 (X, Z) is torsion free. e Let A be the blow up of A in the sixteen fixed points of the involution ι and let πA : A −→ A/ι be the 2 : 1 cover. As in [Mo, Section 3], let HA˜ be the ˜ Z) of the exceptional curves and HKm(A) be the orthogonal complement in H 2 (A, orthogonal complement in H 2 (Km(A), Z) of the 16 (−2)-curves on Km(A). Then HA˜ ∼ = H 2 (A, Z) and there are the natural maps (see [Mo, Section 3]): π ∗ : HKm(A) → H ˜ ∼ = H ˜ → HKm(A) ⊂ H 2 (Km(A), Z) = H 2 (A, Z); πA : H 2 (A, Z) ∼ A

A

A

∗

Lemma 2.5. We have πA∗ (U and so πA ∗ (U ⊕3 ) = U (2)⊕3 .

⊕3

ι∗

∗

) = πA∗ (H (A, Z) ) = H 2 (A, Z)ι (2) = U ⊕3 (2) 2

Proof. Follows from [Mo, Lemma 3.1] and Proposition 2.4. 2 ⊕3 ⊕16 ∼ ⊗ By this lemma we can write ΛK3 ⊗Q = H (Km(A), Q) ≃ U (2) ⊕ < −2 > Q. The lattice U (2)⊕3 ⊕ < −2 >⊕16 has index 211 in ΛK3 ≃ U ⊕3 ⊕ E8 (−1)⊕2 . Proposition 2.6. Let Km(A) be the Kummer surface associated to the Abelian surface A. Then the Picard number of Km(A) is ρ(Km(A)) = ρ(A) + 16, in particular ρ(Km(A)) ≥ 17. The transcendental lattice of Km(A) is TKm(A) = TA (2). The Néron–Severi group ′ N S(Km(A)) is an overlattice KN S(A) of N S(A)(2) ⊕ K and [N S(Km(A)) : (N S(A)(2) ⊕ K)] = 2ρ(A) . Proof. We have that πA∗ (N S(A) ⊕ TA ) = N S(A)(2) ⊕ TA (2) and this lattice is orthogonal to the 16 (−2)-classes in H 2 (Km(A), Z) arising form the desingularization of A/ι. Since πA∗ preserves the Hodge decomposition, we have N S(A)(2) ⊂ N S(Km(A)) and TA (2) = TKm(A) (cf. [Mo, Proposition 3.2]). Hence the Néron–Severi group of Km(A) is an overlattice of finite index of N S(A)(2)⊕K. In fact, rank N S(Km(A)) = 22−rank TA = 22−(6−rank(N S(A))) = 16+rank(N S(A)) = rank(N S(A)(2)⊕K). The index of this inclusion is computed comparing the discriminant of these two lattices indeed 26−ρ(A) d(TA ) = d(TKm(A) ) = d(N S(Km(A))) and d(N S(A)(2) ⊕ K) = 26 2ρ(A) d(N S(A)) = 26+ρ(A) d(TA ), thus d(N S(A)(2)⊕K)/d(N S(Km(A))) = 26+ρ(A) d(TA )/26−ρ(A) d(TA ) = 22ρ(A) which is equal to [N S(Km(A)) : (N S(A)(2)⊕ K)]2 (see e.g. [BHPV, Ch. I, Lemma 2.1]). Now we will consider the generic case, i.e. the case of Kummer surfaces with Picard number 17. By Proposition 2.6, if Km(A) has Picard number 17, then its ′ Néron–Severi group is an overlattice, K4d , of index 2 of N S(A)(2) ⊕ K ≃ ZH ⊕ K 2 where H = 4d, d > 0. In the next proposition we describe the possible overlattices of ZH ⊕ K with H 2 = 4d and hence the possible Néron–Severi groups of the Kummer surfaces with Picard number 17.

6


Theorem 2.7. Let Km(A) be a Kummer surface with Picard number 17, let H be a divisor generating K ⊥ ⊂ N S(Km(A)), H 2 > 0. Let d be a positive integer such ′ ′ that H 2 = 4d and let K4d := ZH ⊕ K. Then: N S(Km(A)) = K4d , where K4d is generated by K4d and by a class (H/2, v4d /2), with: • v4d ∈ K, v4d /2 6∈ K and v4d /2 ∈ K ∨ (in particular v4d · Ki ∈ 2Z); 2 2 • H 2 ≡ −v4d mod 8 (in particular v4d ∈ 4Z). ′ ′ The lattice K4d is the unique even lattice (up to isometry) such that [K4d : K4d ] = 2 ′ and K is a primitive sublattice of K4d . Hence one can assume that: if H 2 ≡ 0 mod 8, then X v4d = Kp = K0,0,0,0 + K1,0,0,0 + K0,1,0,0 + K1,1,0,0 ; p∈V1,2

2

if H ≡ 4 mod 8, then X v4d = Kp = K0,0,0,1 + K0,0,1,0 + K0,0,1,1 + K1,0,0,0 + K0,1,0,0 + K1,1,0,0 . p∈(V1,2 ∗V3,4 )

Proof. The conditions on v4d to construct the lattice K4d can be proved as in ′ [GSa1, Proposition 2.1]. The uniqueness of K4d and the choice of v4d follows from the description of the orbits under the group of isometries of K on the discriminant group K ∨ /K, see Remark 2.3. Remark 2.8. (cf. [G2], [BHPV]) 1) Let ωij := πA∗ (γ ∗ (dxi ∧ dxj )), i < j, i, j = 1, 2, 3, 4 (we use the notation of diagram (1)). The six vectors ωi,j form a basis of U (2)⊕3 . The lattice generated by the Kummer lattice K and by the six classes P uij = 12 (ωij + Ka1 ,a2 ,a3 ,a4 )

where the sum is over (a1 , a2 , a3 , a4 ) ∈ (Z/2Z)4 such that ai = aj = 0, {i, j, h, k} = {1, 2, 3, 4}, and h < k, is isometric to ΛK3 . 2) Observe that since for each d ∈ Z>0 there exist Abelian surfaces with Néron– Severi group isometric to h2di, for each d there exist Kummer surfaces with Néron– ′ Severi group isomorphic to K4d . ′ Let Fd , d ∈ Z>0 denote the family of K4d -polarized K3 surfaces then:

Corollary 2.9. The moduli space of the Kummer surfaces has a countable number of connected components, which are the Fd , d ∈ Z>0 . ′ Proof. Every Kummer surface is polarized with a lattice K4d , for some d, by Propo′ sitions 2.6 and Theorem 2.7. On the other hand if a K3 surface is K4d polarized, then there exists a primitive embedding of K in its Néron–Severi group and by [Ni1, Theorem 1] it is a Kummer surface. P Remark 2.10. The classes of type 21 (H + v4d + p∈W Kp ), where H and v4d ′ are as in Theorem 2.6 and W is a hyperplane of (Z/2Z)4 , are classes in K4d . We describe this kind of classes modulo the lattice ⊕p∈(Z/2Z)4 ZKp . We use the notation of Theorem 2.7. ′ If H 2 = 4d ≡ 0 mod 8, the lattice K4d contains: P 1 • 4 classes of type 2 (H − p∈J4 Kp ) for certain J4 ⊂ (Z/2Z)4 which contain 4 P elements: these classes are 21 (H +v4d ) and the classes 12 (H +v4d + p∈W Kp ) where W ⊃ {(0, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0), (1, 1, 0, 0)} ;


7

P • 24 classes of type 21 (H − p∈J8 Kp ) for certain J8 ⊂ (Z/2Z)4 which contain P 8 elements: these classes are 12 (H + v4d + p∈W Kp ) where W ∩ {(0, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0), (1, P 1, 0, 0)} contains 2 elements. • 4 classes of type 21 (H − p∈J12 Kp ) for certain J12 ⊂ (Z/2Z)4 which conP tain 12 elements: these classes are 12 (H + v4d + p∈(Z/2Z)4 Kp ) and the P classes 21 (H + v4d + p∈W Kp ) where W ∩ {(0, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0), (1, 1, 0, 0)} = ∅. ′ If H 2 = 4d ≡ 4 mod 8, the lattice K4d contains: P 1 • 16 classes of type 2 (H − p∈J6 Kp ) for certain J6 ⊂ (Z/2Z)4 which con1 1 tain P six elements: these classes are 2 (H + v4d ) and the classes 2 (H + v4d + p∈W Kp ) where W ∩{(1, 0, 0, 0), (0, 1, 0, 0), (1, 1, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0), (0, 0, 1, 1)} contains 4 elements; P • 16 classes of type 21 (H − p∈J10 Kp ) for certain J10 ⊂ (Z/2Z)4 which P contain 10 elements: these classes are 12 (H + v4d + p∈(Z/2Z)4 Kp ) and the P classes 12 (H + v4d + p∈W Kp ) where W ∩ {(1, 0, 0, 0), (0, 1, 0, 0), (1, 1, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0), (0, 0, 1, 1)} contains 2 elements. P ′ Remark 2.11. The discriminant group of K4d is generated by (H/4d)+ 12 ( p∈V3,4 Kp ), P P P P 1 1 1 1 2 = 4d ≡ 0 p∈V1,3 Kp ), 2 ( p∈V1,4 Kp ), 2 ( p∈V2,3 Kp ), 2 ( p∈V2,4 Kp ) if H 2( P P P P 1 1 1 1 mod 8 and by (H/4d)+ 2 ( p∈V1,2 Kp ), 2 ( p∈V1,3 Kp ), 2 ( p∈V1,4 Kp ), 2 ( p∈V2,3 Kp ), P 1 2 p∈V2,4 Kp ) if H = 4d ≡ 4 mod 8. 2( 2.2. Kummer surfaces as K3 surfaces with 16 nodes. Let S be a surface with n nodes and let Se be its minimal resolution. On Se there are n disjoint rational curves which arise from the resolution of the nodes of S. If Se is a K3 surface, then n ≤ 16, [Ni1, Corollary 1]. By [Ni1, Theorem 1], if a K3 surface admits 16 disjoint rational curves, then they form an even set and the K3 surface is in fact a Kummer surface. Conversely, as remarked in the previous section, every Kummer surface contains 16 disjoint rational curves. Thus, the Kummer surfaces are the K3 surfaces admitting the maximal numbers of disjoint rational curves or equivalently they are the K3 surfaces which admit a singular model with the maximal number of nodes. 2.3. Kummer surfaces as quotient of K3 surfaces. Definition 2.12. (cf. [Mo, Definition 5.1]) An involution ι on a K3 surface Y is a Nikulin involution if ι∗ ω = ω for every ω ∈ H 2,0 (Y ). Every Nikulin involution has eight isolated fixed points and the minimal resolution X of the quotient Y /ι is again a K3 surface ([Ni3, §11, Section 5]). The minimal primitive sublattice of N S(X) containing the eight exceptional curves from the resolution of the singularities of Y /ι is called Nikulin lattice and it is denoted by N , its discriminant is 26 . Definition 2.13. (cf. [vGS]) A Nikulin involution ι on a K3 surface Y is a Morrison–Nikulin involution if ι∗ switches two orthogonal copies of E8 (−1) embedded in N S(Y ). By definition, if Y admits a Morrison–Nikulin involution then E8 (−1)⊕E8 (−1) ⊂ N S(Y ). A Morrison–Nikulin involution has the following properties (cf. [Mo, Theorem 5.7 and 6.3]):

8


• TX = TY (2); • the lattice N ⊕ E8 (−1) is primitively embedded in N S(X); • the lattice K is primitively embedded in N S(X) and so X is a Kummer surface. Definition 2.14. (cf. [Mo, Definition 6.1]) Let Y be a K3 surface and ι be a Nikulin involution on Y . The pair (Y, ι) is a Shioda–Inose structure if the rational quotient map π : Y _ _ _/ X is such that X is a Kummer surface and π∗ induces a Hodge isometry TY (2) ∼ = TX . The situation is resumed in the following diagram (A0 denotes an abelian surface): zz zz z z z }z 0 A /i o

Y A A0 L AA LL ss AA s LL s AA s A L% ys / Y /ι X = Km(A0 )

We have TY ∼ = TA0 by [Mo, Theorem 6.3]. Let Y be a K3 surface and ι be a Nikulin involution on Y . By [Mo, Theorem 5.7 and 6.3] we conclude that Corollary 2.15. A pair (Y, ι) is a Shioda–Inose structure if and only if ι is a Morrison–Nikulin involution. For the next result see [OS, Lemma 2]. Proposition 2.16. Every Kummer surface is the desingularization of the quotient of a K3 surface by a Morrison–Nikulin involution, i.e. it is associated to a Shioda– Inose structure. Remark 2.17. In [OS, Lemma 5] it is proved that if X is a K3 surface with ρ(X) = 20, then each Shioda–Inose structure is induced by the same Abelian surface. This means that if (X, ι1 ) and (X, ι2 ) are Shioda–Inose structures and Yi = Km(Bi ) is the Kummer surface minimal resolution of Xi /ιi , i = 1, 2 then B1 = B2 , and so Y1 = Y2 . By Proposition 2.16 it follows that Kummer surfaces can be defined also as K3 surfaces which are desingularizations of the quotients of K3 surfaces by Morrison– Nikulin involutions. This definition leads to a different description of the Néron– Severi group of a Kummer surface, which we give in the following: Theorem 2.18. Let Y be a K3 surface admitting a Morrison–Nikulin involution ι, then ρ(Y ) ≥ 17 and N S(Y ) ≃ R ⊕ E8 (−1)2 where R is an even lattice with signature (1, ρ(Y ) − 17). Let X be the desingularization of Y /ι, then N S(X) is an overlattice of index 2(rank(R)) of R(2) ⊕ N ⊕ E8 (−1). In particular, if ρ(Y ) = 17, then: N S(Y ) ≃ h2di ⊕ E8 (−1)2 , the surface X is the Kummer surface of the (1, d)-polarized Abelian surface and the Néron–Severi group of X is an overlattice of index 2 of h4di ⊕ N ⊕ E8 (−1). Proof. By [Mo, Theorem 6.3] and the fact that E8 (−1) is unimodular one can write N S(Y ) = R ⊕ E8 (−1)⊕2 , with R even of signature (1, ρ(Y ) − 17). In [Mo, Theorem 5.7] it is proved that N ⊕E8 (−1) is primitively embedded in N S(X). Thus, arguing on the discriminant of the transcendental lattices of Y and X and on the lattice R


9

as in Proposition 2.6, one concludes the first part of the proof. For the last part of the assertion observe that the lattices N S(Y ) and TY are uniquely determined by their signature and discriminant form ([Mo, Theorem 2.2]), so TY = h−2di ⊕ U 2 . By construction TY (2) = TX = TA0 (2) so TA0 = TY . This determines uniquely N S(A0 ), which is isometric to h2di. Hence A0 is a (1, d)-polarized abelian surface. ′ The overlattices N2d of index 2 of h2di ⊕ N are described in [GSa1] and, by ′ Theorem 2.18, we conclude that if ρ(Y ) = 17 then N S(X) ≃ N4d ⊕ E8 (−1). Remark 2.19. Examples of Shioda–Inose structures on K3 surfaces with Picard number 17 are given e.g. in the appendix of [GaLo], in [Kum1], [vGS], [Koi] and in [Sc]. In all these papers the Morrison–Nikulin involutions of the Shioda–Inose structures are induced by a translation by a 2-torsion section on an elliptic fibration. In particular, in [Koi] all the Morrison–Nikulin involutions induced in such a way on elliptic fibrations with a finite Mordell–Weil group are classified. Remark 2.20. Proposition 2.6 and Theorem 2.18 give two different descriptions of the same lattice (the Néron–Severi group of a Kummer surface of Picard number 17): the first one is associated to the construction of the Kummer surface as quotient of an Abelian surface; the second one is associated to the construction of the same surface as quotient of another K3 surface. In general it is an open problem to pass from one description to the other, and hence to find the relation among these two constructions of a Kummer surface. However in certain cases this relation is known. In [Na] Naruki describes the Néron–Severi group of the Kummer surface of the Jacobian of a curve of genus 2 as in our Proposition 2.6 and he determines a nef divisor that gives a 2 : 1 map to P2 (we describe this map in Section 5.1). Then, 16 curves on P2 are constructed and it is proved that their pull backs on the Kummer surface generate the lattice N ⊕ E8 (−1). Similarly this relation is known if the Abelian surface is E × E ′ , the product of two non isogenous elliptic curves E, E ′ . In [O] the Néron–Severi group of Km(E × E ′ ) is described as in Proposition 2.6. Then the elliptic fibrations on this K3 surface are classified. In particular there exists an elliptic fibration with a fiber of type II ∗ and two fibers of type I0∗ . The components of II ∗ which do not intersect the zero section generate a lattice isometric to E8 (−1) and are orthogonal to the components of I0∗ . The components with multiplicity 1 of the two fibers of type I0∗ generate a lattice isometric to N and orthogonal to the copy of E8 (−1) that we have described before. Thus, one has an explicit relation between the two descriptions of the Néron–Severi group. 3. Automorphisms on Kummer surfaces It is in general a difficult problem to describe the full automorphisms group of a given K3 surface. However for certain Kummer surfaces it is known. For example the group of automorphisms of the Kummer surface of the Jacobian of a curve of genus 2 is described in [Ke1] and [Kon]. Similarly the group Aut(Km(E × F )) is determined in [KK] in the cases: E and F generic and non isogenous, E and F generic and isogenous, E and F isogenous and with complex multiplication. A different approach to the study of the automorphisms of K3 surfaces is to fix a particular group of automorphisms and to describe the families of K3 surfaces admitting such (sub)group of automorphisms. For this point of view the following two known results (Propositions 3.1 and 3.3) which assure that every Kummer

10


surface admits some particular automorphisms are important. Moreover, we prove also a result (Proposition 3.5) which limits the list of the admissible finite group of symplectic automorphisms on a generic Kummer surface. 3.1. Enriques involutions on Kummer surfaces. We recall that an Enriques involution is a fixed point free involution on a K3 surface. Proposition 3.1. ([Ke2, Theorem 2]) Every Kummer surface admits an Enriques involution. To prove the proposition, in [Ke2] the following is shown first (see [Ni2] and [Ho]). Proposition 3.2. ([Ke2, Theorem 1]) A K3 surface admits an Enriques involution if and only if there exists a primitive embedding of the transcendental lattice of the surface in U ⊕ U (2) ⊕ E8 (−2) such that its orthogonal complement does not contain classes with self–intersection equal to −2. In [Ke2], the author applies the proposition to the transcendental lattice of any Kummer surface. We observe that this does not give an explicit geometric description of the Enriques involution. 3.2. Finite groups of symplectic automorphisms on Kummer surfaces. Proposition 3.3. (see e.g.[G2]) Every Kummer surface Km(A) admits G = (Z/2Z)4 as group of symplectic automorphisms. These are induced by the translation by points of order two on the Abelian surface A and the desingularization of Km(A)/G is isomorphic to Km(A) (thus every Kummer surface is also the desingularization of the quotient of a Kummer surface by (Z/2Z)4 ). Proof. Let A[2] be the group generated by 2-torsion points. This is isomorphic with (Z/2Z)4 , it operates on A by translation and commutes with the involution ι. Hence it induces an action of G = (Z/2Z)4 on Km(A), and so on H 2 (Km(A), Z). Observe that G leaves the lattice U (2)⊕3 ≃ hωij i invariant, in fact G as a group generated by translation on A does not change the real two forms dxi ∧ dxj . Since TKm(A) ⊂ U (2)⊕3 the automorphisms induced on Km(A) by G are symplectic. Moreover since ι and G commute we obtain that the surface Km(A/A[2]) and ·2 ^ are isomorphic. Finally from the exact sequence 0 → A[2] → A → Km(A)/G A→ ^ ≃ Km(A/A[2]) ≃ Km(A). 0 we have A/A[2] ∼ = A and so Km(A)/G

Remark 3.4. One can also consider the quotient of Km(A) by subgroups of G = (Z/2Z)4 , for example by one involution. Such an involution is induced by the translation by a point of order two. Take the Abelian surface A ∼ = R4 /Λ, where Λ = h2e1 , e2 , e3 , e4 i and consider the translation te1 by e1 . Thus, A/hte1 i is the Abelian surface B := R4 /he1 , e2 , e3 , e4 i. So the desingularization of the quotient of Km(A) by the automorphism induced by te1 is again a Kummer surface and more precisely it is Km(B). In particular if N S(A) = h2di, then N S(B) = h4di, [BL]. ′ ′ This implies that if N S(Km(A)) ≃ K4d , then N S(Km(B)) ≃ K8d . Analogously n one can consider the subgroups Gn = (Z/2Z) ⊂ G (generated by translations), ′ ′ n = 1, 2, 3: if N S(Km(A)) ≃ K4d , then N S(Km(A/Gn )) ≃ K4·2 n ·d . Proposition 3.5. Let G be a finite group of symplectic automorphisms of a Kummer surface Km(A), where A is a (1, d)-polarized Abelian surface and ρ(A) = 1. Then G is either a subgroup of (Z/2Z)4 , or Z/3Z or Z/4Z.


11

Proof. Let G be a finite group acting symplectically on a K3 surface and denote by ΩG the orthogonal complement of the G-invariant sublattice of the K3 lattice ΛK3 . An algebraic K3 surface admits the group G of symplectic automorphisms if and only if ΩG is primitively embedded in the Néron–Severi group of the K3 surface (cf. [Ni3], [Ha]), hence the Picard number is greater than or equal to rank(ΩG )+1. The list of the finite groups acting symplectically on a K3 surface and the values of rank(ΩG ) can be found in [X, Table 2] (observe that Xiao considers the lattice generated by the exceptional curves in the minimal resolution of the quotient, he denotes its rank by c. This is the same as rank(ΩG ) by [I, Corollary 1.2]). Since we are considering Kummer surfaces such that ρ(Km(A)) = 17, if G acts symplectically, then rank(ΩG ) ≤ 16. This gives the following list of admissible groups G: (Z/2Z)i for i = 1, 2, 3, 4, Z/nZ for n = 3, 4, 5, 6, Dm for m = 3, 4, 5, 6, where Dm is the dihedral group of order 2m, Z/2Z × Z/4Z, (Z/3Z)2 , Z/2Z × D4 , A3,3 (see [Mu] for the definition), A4 . We can exclude that G acts symplectically on a Kummer surface for all the listed cases except (Z/2Z)i , i = 1, 2, 3, 4, Z/3Z and Z/4Z by considering the rank and the length of the lattice ΩG , which is the minimal number of generators of the discriminant group. For example, let us consider the case G = D3 . The lattice ΩD3 is an even negative definite lattice of rank 14. Since the group D3 can be generated by two involutions, ΩD3 is the sum of two (non orthognal) copies of ΩZ/2Z ≃ E8 (−2) and admits D3 as group of isometries (cf. [G3, Remark 7.9]). In fact ΩD3 ≃ DIH6 (14) where DIH6 (14) is the lattice described in [GrLa, Section 6]. The discriminant group of ΩD3 ≃ DIH6 (14) is (Z/3Z)3 × (Z/6Z)2 , [GrLa, Table 8]. If D3 acts symplectically on Km(A), N S(Km(A)) is an overlattice of finite index of ΩD3 ⊕ R where R is a lattice of rank 3. But there are no overlattices of finite index of ΩD3 ⊕ R with discriminant group (Z/2Z)4 × Z/2dZ, which is the discriminant group of N S(Km(A)). Indeed, for every overlattice of finite index of ΩD3 ⊕ R , since the rank of R is 3, the discriminant group contains at least two copies of Z/3Z. In order to exclude all the other groups G listed before, one has to know the rank and the discriminant group of ΩG : this can be found in [GSa2, Proposition 5.1] if G is abelian; in [G3, Propositions 7.6 and 8.1] if G = Dm , m = 4, 5, 6, G = Z/2Z × D4 and G = A3,3 ; in [BG, Section 4.1.1] if G = A4 . Remark 3.6. We can not exclude the presence of symplectic automorphisms of order 3 or 4 on a Kummer surface with Picard number 17, but we have no explicit examples of such an automorphism. It is known that there are no automorphisms of such type on Km(A), if A is principally polarized, cf. [Ke1], [Kon]. If Km(A) admits a symplectic action of Z/3Z, then d ≡ 0 mod 3 (this follows comparing the length of ΩZ/3Z and of N S(Km(A)) as in the proof of Proposition 3.5). Moreover, the automorphism of order 3 generates an infinite group of automorphisms with any symplectic involution on Km(A). Otherwise, if they generate a finite group, it has to be one of the groups listed in Proposition 3.5, but there are no groups in this list containing both an element of order 2 and one of order 3. 3.3. Morrison-Nikulin involutions on Kummer surfaces. Examples of certain symplectic automorphisms on a Kummer surface (the Morrison-Nikulin involutions) come from the Shioda–Inose structure. We recall that every K3 surface with Picard number at least 19 admits a Morrison–Nikulin involution. In particular this holds true for Kummer surfaces of Picard number at least 19. This is false for Kummer surfaces with lower Picard number. In fact since a Kummer surface

12


with a Morrison–Nikulin involution admits also a Shioda–Inose structure as shown in Section 2.3 it suffices to prove: Corollary 3.7. Let Y ∼ = Km(B) be a Kummer surface of Picard number 17 or 18, then Y does not admit a Shioda–Inose structure. Proof. If a K3 surface Y admits a Shioda–Inose structure, then by Theorem 2.18 we can write N S(Y ) = R ⊕ E8 (−1)2 with R an even lattice of rank 1 or 2, hence the length of N S(Y ) satisfies l(AN S(Y ) ) ≤ 2. It follows immediately that we have also l(ATY ) ≤ 2. Let e1 , . . . , ei , i = 5 respectively 4, be the generators of TY . Since Y ∼ = Km(B) we have that TY = TB (2) and so the classes ei /2 are independent elements of TY∨ /TY thus we have 2 ≥ l(ATY ) ≥ 4, which is a contradiction. In the case the Picard number is 19 we can give a more precise description of the Shioda–Inose structure: Proposition 3.8. Let Y ≃ Km(B) be a Kummer surface, ρ(Y ) = 19 (so Y admits a Morrison–Nikulin involution ι). Let Km(A0 ) be the Kummer surface which is the desingularization of Y /ι. Then A0 is not a product of two elliptic curves. Proof. If A0 = E1 × E2 , Ei , i = 1, 2 an elliptic curve, then the classes of E1 and E2 in N S(A0 ) span a lattice isometric to U . To prove that A0 is not such a product it suffices to prove that there is not a primitive embedding of U in N S(A0 ). Assume the contrary, then N S(A0 ) = U ⊕ Zh, so ℓ(N S(A0 )) = 1. Since Y ≃ Km(B) is a Kummer surface, TY ≃ TB (2) and thus TA0 ≃ TB (2), this implies that 1 = ℓ(N S(A0 )) = ℓ(TA0 ) = 3 which is a contradiction. 4. Ampleness of divisors on Kummer surfaces In this section we consider projective models of Kummer surfaces with Picard number 17. The main idea is that we can check if a divisor is ample, nef, or big and nef (which is equivalent to pseudo ample) because we have a complete description of the Néron–Severi group and so of the (−2)-curves. Hence we can apply the following criterion (see [BHPV, Proposition 3.7]): Let L be a divisor on a K3 surface such that L2 ≥ 0, then it is nef if and only if L · D ≥ 0 for all effective divisors D such that D2 = −2. This idea was used in [GSa1, Proposition 3.2], where one proves that if there exists a divisor with a negative intersection with L then this divisor has selfintersection strictly less than −2. We refer to the description of the Néron–Severi group given in Proposition 2.6, where the Néron–Severi group is generated, over Q, by an ample class and by 16 disjoint rational curves, which form an even set over Z. Since the proofs of the next propositions are very similar to the ones given in [GSa1, Section 3] (where the Néron–Severi groups of the K3 surfaces considered are generated over Q by an ample class and by 8 disjoint rational curves forming an even set) we omit them. We denote by φL the map induced by the ample (or nef, or big and nef) divisor L on Km(A). Proposition 4.1. (cf. [GSa1, Proposition 3.1]) Let Km(A) be a Kummer surface ′ such that N S(Km(A)) ≃ K4d . Let H be as in Theorem 2.7. Then we may assume that H is pseudo ample and |H| has no fixed components.


13

Remark 4.2. The divisor H is orthogonal to all the curves of the Kummer lattice, so φH contracts them. The projective model associated to this divisor is an algebraic K3 surface with sixteen nodes forming an even set. More precisely φH (Km(A)) is a model of A/ι. Proposition 4.3. (cf. [GSa1, Propositions 3.2 and 3.3]) Let Km(A) be a Kummer ′ surface such that N S(Km(A)) ≃ K4d . P • If d ≥ 3, i.e. H 2 ≥ 12, then the class H − 12 ( p∈(Z/2Z)4 Kp ) ⊂ N S(Km(A)) P P is an ample class. Moreover m(H− 12 ( p∈(Z/2Z)4 Kp )) and mH− 21 ( p∈(Z/2Z)4 Kp ) for m ∈ Z>0 , are ample. P P • If d = 2, i.e. (H − 12 ( p∈(Z/2Z)4 Kp ))2 = 0, then m(H − 21 ( p∈(Z/2Z)4 Kp )) P is nef for m ≥ 1 and mH − 21 ( p∈(Z/2Z)4 Kp ) is ample for m ≥ 2. P Proposition 4.4. (cf. [GSa1, Proposition 3.4]) The divisors H− 12 ( p∈(Z/2Z)4 Kp ), P P mH − 21 ( p∈(Z/2Z)4 Kp ) and m(H − 12 ( p∈(Z/2Z)4 Kp )), m ∈ Z>0 , do not have fixed components for d ≥ 2. Lemma 4.5. (cf. [GSa1, Lemma 3.1]) The map φH− 12 (Pp∈(Z/2Z)4 Kp ) is an embedding if H 2 ≥ 12. Proposition 4.6. (cf. [GSa1, Proposition 3.5]) 1) Let D be the divisor D = H − (K1 + . . . + Kr ) (up to relabelling of the indices), 1 ≤ r ≤ 16. Then D is pseudo ample for 2d > r. ¯ = (H − K1 − . . . − Kr )/2 with r = 4, 8, 12 if d ≡ 0 mod 2 and 2) Let D r = 6, 10 if d ≡ 1 mod 2. Then: ¯ is pseudo ample whenever it has positive self-intersection, • the divisor D ¯ is pseudo ample then it does not have fixed components, • if D ¯ 2 = 0 then the generic element in |D| ¯ is an elliptic curve. • if D P Remark 4.7. In the assumptions of Lemma 4.5 the divisor H − 12 ( p∈(Z/2Z)4 Kp ) defines an embedding of the surface Km(A) into a projective space which sends the curves of the Kummer lattice to lines. A divisor D as in Proposition 4.6 defines a map from the surface Km(A) to a projective space which contracts some rational ¯ curves of the even set and sends the others to conics on the image. Similarly, D defines a map from the surface Km(A) to a projective space which contracts some rational curves of the even set and sends the others to lines on the image. 5. Projective models of Kummer surfaces with Picard number 17 Here we consider certain Kummer surfaces with Picard number 17 and we describe projective models determined by the divisors presented in the previous section. Some of these models (but not all) are very classical. 5.1. Kummer of the Jacobian of a genus 2 curve. Let C be a general curve of genus 2. It is well known that the Jacobian J(C) is an Abelian surface such that N S(J(C)) = ZL, with L2 = 2 and TJ(C) ≃ h−2i⊕U ⊕U . Hence N S(Km(J(C))) ≃ K4′ and TKm(J(C)) ≃ h−4i ⊕ U (2) ⊕ U (2) (see Proposition 2.6 and Theorem 2.7). Here we want to reconsider some known projective models of Km(J(C)) (see [GH, Chapter 6]) using the description of the classes in the Néron–Severi group introduced in the previous section.

14


The singular quotient surface J(C)/ι is a quartic in P3 with sixteen nodes. For each of these nodes there exist six planes which pass through that node and each plane contains other five nodes. Each plane cuts the singular quartic surface in a conic with multiplicity 2. In this way we obtain 16 hyperplane sections which are double conics. These 16 conics are called tropes. They are the image, under the quotient map J(C) → J(C)/ι of different embeddings of C in J(C). We saw that every Kummer surface admits an Enriques involution (cf. Proposition 3.1). If the Kummer surface is associated to the Jacobian of a curve a genus 2, an explicit equation of this involution on the singular model of Km(J(C)) in P3 is given in [Ke2, Section 3.3]. The polarization H. The map φH contracts all the curves in the Kummer lattices and hence φH (Km(J(C))) is the singular quotient J(C)/ι in P3 . The class H is the image in N S(Km(J(C))) of the class generating N S(J(C)) (Proposition 2.6). The classes corresponding to the tropes are the 16 classes P (described in Remark 2.10, case 4d ≡ 4 mod 8) of the form uJ6 := 21 (H − p∈J6 Kp ). Indeed P 2uJ6 + p∈J6 Kp = H so they correspond to a curve in a hyperplane section with multiplicity 2; u2J6 = −2, so they are rational curves; uJ6 · H = 2, so they have degree 2. In particular the trope corresponding to the class uJ6 passes through the nodes obtained by contracting the six curves Kp , where p ∈ J6 . It is a classical result (cf. [Hud, Ch. I, §3]) that the rational curves of the Kummer lattice and the rational curves corresponding to the tropes in this projective models form a 166 configuration of rational curves on Km(J(C)). This can directly checked considering the intersections between the curves Kp , p ∈ (Z/2Z)4 and the classes u J6 . The polarization H − K0,0,0,0 . Another well known model is obtained projecting the quartic surface in P3 from a node. This gives a 2 : 1 cover of P2 branched along six lines which are the image of the tropes passing through the node from which we are projecting. The lines are all tangent to a conic (cf. [Na, §1]). Take the node associated to the contraction of the curve K0,0,0,0 then the linear system associated to the projection of J(C)/ι from this node is |H − K0,0,0,0 |. The classes uJ6 such that (0, 0, 0, 0) ∈ J6 are sent to lines and the curve K0,0,0,0 is sent to a conic by the map φH−K0,0,0,0 : Km(J(C)) → P2 . This conic is tangent to the lines which are the images of the tropes uJ6 . So the map φH−K0,0,0,0 : Km(J(C)) → P2 exhibits Km(A) as double cover of P2 branched along six lines tangent to the conic C := φH−K0,0,0,0 (K0,0,0,0 ). The singular points of the quartic J(C)/ι which are not the center of this projection are singular points of the double cover of P2 . So the classes Ka1 ,a2 ,a3 ,a4 of the Kummer lattice such that (a1 , a2 , a3 , a4 ) 6= (0, 0, 0, 0) are singular points for φH−K0,0,0,0 (Km(J(C))) and in fact correspond to the fifteen intersection points of the six lines in the branch locus. Observe that if one fixes three of the six lines, the conic C is tangent to the edges of this triangle. The remaining three lines form a triangle too and the edges are tangent to the conic C. By a classical theorem of projective plane geometry (a consequence of Steiner’s theorem on generation of conics) the six vertices of the triangles are contained in another conic D, and in fact this conic is the image of one of the tropes which do not pass through the singular point corresponding to K0,0,0,0 . This can be checked directly on N S(Km(J(C))). Observe that we have in total 10 such conics. Deformation. We observe that this model of Km(J(C)) exhibits the surface as a special member of the 4-dimensional family of K3 surfaces which are 2 : 1 cover of


15

P2 branched along six lines in general position. The covering involution induces a non-symplectic involution on Km(J(C)) which fixes 6 rational curves. By Nikulin’s classification of non-symplectic involutions (cf. e.g. [AN, Section 2.3]) the general member of the family has Néron–Severi group isometric to h2i ⊕ A1 ⊕ D4 ⊕ D10 and transcendental lattice isometric to U (2)⊕2 ⊕ h−2i⊕2 which clearly contains TKm(J(C)) ≃ U (2)⊕2 ⊕ h−2i. This is a particular case of Proposition 7.13. P The polarization 2H − 12 p∈(Z/2Z)4 Kp . We denote by D this polarization. The divisor D is ample by Proposition 4.3. Since D2 = 8 the map φD gives a smooth projective model of Km(J(C)) as intersection of 3 quadrics in P5 . Using suitable coordinates, we can write C as y2 =

5 Y

(x − si )

i=0

with si ∈ C, si 6= sj for i 6= j (it is the double cover of P1 ramified on six points). Then by [Sh2, Theorem 2.5], φD (Km(J(C))) has equation  2  z0 + z12 + z22 + z32 + z42 + z52 = 0 s0 z 2 + s1 z12 + s2 z22 + s3 z32 + s4 z42 + s5 z52 = 0 (2)  2 02 s0 z0 + s21 z12 + s22 z22 + s23 z32 + s24 z42 + s25 z52 = 0

in P5 . The curves of the Kummer lattice are sent to lines by the map φD , indeed D · Kp = 1 for each p ∈ (Z/2Z)4 . The image of the rational curves associated to a divisor of type uJ6 (i.e. the curves which are tropes on the surface φH (Km(J(C)))) are lines: in fact one computes D · uJ6 = 1. So on the surface φD (Km(J(C))) we have 32 lines which admits a 166 configuration. Keum [Ke1, Lemma 3.1] proves that the set of the tropes and the curves Kp , p ∈ (Z/2Z)4 , generate the Néron–Severi group (over Z). Here we find the same result as a trivial application of Theorem 2.7. Moreover we can give a geometric interpretation of this fact, indeed this implies that the Néron–Severi group of the surface φD (Km(J(C))) is generated by lines (other results about the Néron–Severi group of K3 surfaces generated by lines can be found e.g. in [BS]). More precisely the following hold: Proposition 5.1. The Néron–Severi group of the K3 surfaces which are smooth complete intersections of the three quadrics in P5 defined by (2) is generated by lines. Proof. With the help of Theorem 2.7 we find here a set of classes generating N S(Km(J(C))) which correspond to lines in the projective model of the Kummer surface φD (Km(J(C))). This set of classes is S := { e1 := 21 (H − v4 ), e2 := 1 1 2 (H−K0,0,0,0 −K1,0,0,0 −K0,1,0,1 −K0,1,1,0 −K1,1,0,0 −K0,1,1,1 ), e3 := 2 (H−K0,0,0,0 − 1 K0,1,0,0 − K1,1,0,0 − K1,0,1,0 − K1,0,0,1 − K1,0,1,1 ), e4 := 2 (H − K0,0,0,0 − K0,0,1,0 − K0,0,1,1 − K1,0,0,1 − K0,1,0,1 − K1,1,0,1 ), e5 := 21 (H − K0,0,0,0 − K0,0,0,1 − K0,0,1,1 − K1,0,1,0 − K1,1,1,0 − K0,1,1,0 ), e6 := 21 (H − K0,0,0,0 − K1,0,0,0 − K0,1,0,0 − K1,1,0,1 − K1,1,1,0 − K1,1,1,1 ), K0,0,0,0 , K1,0,0,0 , K0,1,0,0 , K0,0,1,0 , K0,0,0,1 , K0,0,1,1 , K0,1,0,1 , K1,0,0,1 , K0,1,1,0 , K1,0,1,0 , K1,1,0,0 }. Indeed, by Theorem 2.7, a set of generators of N S(Km(J(C))) is given by e1 and a set of generators of the Kummer lattice K (a set of P generators of K is described in Remark 2.10). Since for j = 2, 3, P4, 5 ej − e1 ≡ (1/2) p∈Wj−1 Kp mod (⊕p∈(Z/2Z)4 ZKp ) and e1 − e2 + e3 − e6 ≡ 21 p∈(Z/2Z)4 Kp mod (⊕p∈(Z/2Z)4 ZKp ), S is a Z-basis of N S(Km(J(C))). It is immediate to check

16


that every element of this basis has intersection 1 with D and thus is sent to a line by φD . P The nef class H − 21 (H − p∈Wi Kp ). Without loss of generality we consider ¯ By Proposition 4.6, it defines an elliptic fibration on i = 1 and we call this class D. ¯ are sections of the Mordell– Km(J(C)) and the eight (−2)-classes contained in D Weil group, the others eight (−2)-classes are components of the reducible fibers. Observe that the class 21 (H−K1,0,0,0 −K1,1,0,0 −K0,1,0,1 −K0,1,1,0 −K0,1,1,1 −K0,0,0,0 ) ¯ and meets the classes K1,0,0,0 has self intersection −2, has intersection 0 with D and K1,1,0,0 in one point. One can find easily 3 classes more as the previous one, so that the fibration contains 4 fibers I4 . Checking in [Kum2, Table p. 9] one sees that this is the fibration number 7 so it has no more reducible fibers and the rank of the Mordell–Weil group is 3. Shioda–Inose structure. We now describe the 3-dimensional family of K3 surfaces which admit a Shioda–Inose structure associated to Km(J(C)) as described in Theorem 2.18. It is obtained by considering K3 surfaces X with ρ(X) = 17 ∗ and with an elliptic fibration with reducible fibers I10 + I2 and Mordell-Weil group equal to Z/2Z (see Shimada’s list of elliptic K3 surfaces [Shim, Table 1, nr. 1343] on the arXiv version of the paper). By using the Shioda-Tate formula (cf e.g. [Sh1, Corollary 1.7]) the discriminant of the Néron–Severi group of such a surface is (22 · 2)/22 . The translation t by the section of order 2 on X is a Morrison– Nikulin involution, indeed it switches two orthogonal copies of E8 (−1) ⊂ N S(X). Thus, the Néron–Severi group is h2di ⊕ E8 (−1) ⊕ E8 (−1), and d = 1 because the discriminant is 2. Hence X has a Shioda–Inose structure associated to the Abelian surface J(C). The desingularization of the quotient X/t is the Kummer surface Km(J(C)) and has an elliptic fibration induced by the one on X, with reducible fibers I5∗ + 6I2 (this is the number 23 of [Kum2]) and Z/2Z as Mordell-Weil group. This Shioda–Inose structure was described in [Kum1, Section 5.3]. In Theorem 2.18 we gave a description of the Néron–Severi group of Km(J(C)) related to the Shioda–Inose structure. In particular we showed that N S(Km(J(C)) is an overlattice of index 2 of h4i ⊕ N ⊕ E8 (−1). We denote by Q the generator of h4i, by Ni i = 1, . . . , 8 the classes of the rational curves in the Nikulin lattice N and by Ej , j = 1, . . . , 8 the generators of E8 (−1) (we assume that Ej , j = 1, . . . , 7 generate a copy of A7 (−1) and E P3 · E8 = 1). Then a Z-basis of N S(Km(J(C))) is {(Q + N1 + N2 ) /2, N1 , . . . , N7 , 8 Ni /2, E1 , . . . , E8 }. It is i=1 easy to identify a copy of N and an orthogonal copy of E8 (−1) in the previous elliptic fibration (the one with reducible fibers I5∗ + I6 ); in particular one remarks that the curves Ni and Ej , j = 2, . . . , 8 are components of the reducible fibers and the curve E1 can be chosen to be the zero section. This immediately gives the class of the fiber in terms of the previous basis of the Néron–Severi group: F := Q − 4E1 − 7E2 − 10E3 − 8E4 − 6E5 − 4E6 − 2E7 − 5E8 . 5.2. Kummer surface of a (1, 2)-polarized Abelian surface. In this section A will denote always a (1, 2) polarized Abelian surface, and N S(A) = ZL where L2 = 4. The polarization H. By Proposition 4.1 the divisor H is pseudo-ample and the singular model φH (Km(A)) has sixteen singular points (it is in fact A/ι). Since H 2 = 8 and since by [SD, Theorem 5.2] H is not hyperelliptic, the K3 surface


17

φH (Km(A)) is a complete intersection of three quadrics in P5 . This model is described by Barth in [Ba1]: Proposition 5.2. [Ba1, Proposition 4.6] Let us consider the following quadrics: Q1 = {(µ21 + λ21 )(x21 + x22 ) − 2µ1 λ1 (x23 + x24 ) + (µ21 − λ21 )(x25 + x26 ) = 0} Q2 = {(µ22 + λ22 )(x21 − x22 ) − 2µ2 λ2 (x23 − x24 ) + (µ22 − λ22 )(x25 − x26 ) = 0} Q3 = {(µ23 + λ23 )x1 x2 − 2µ3 λ3 x3 x4 + (µ23 − λ23 )x5 x6 = 0}. Let r = r1,2 r2,3 r3,1 where rk,j = (λ2j µ2k − λ2k µ2j )(λ2j λ2k − µ2k µ2j ). If r 6= 0 the quadrics Q1 , Q2 , Q3 , generate the ideal of an irreducible surface Q1 ∩ Q2 ∩ Q3 ⊂ P5 of degree 8, which is smooth except for 16 ordinary double points and which is isomorphic to A/ι. The surface A/ι is then contained in each quadric of the net: α1 Q1 +α2 Q2 +α3 Q3 , αi ∈ C. We observe that the matrix M associated to this net of quadrics is a block matrix  B1 0 0 α1 (µ21 + λ21 ) + α2 (µ22 + λ22 ) B2 0  , where B1 = M = 0 α3 (µ23 + λ23 ) 0 0 B 3 −2α1 µ1 λ1 − 2α2 µ2 λ2 −2α3 µ3 λ3 B2 = , −2α1 µ1 λ1 + 2α2 µ2 λ2 −2α32µ3 λ32 2 2 2 2 α1 (µ1 − λ1 ) + α2 (µ2 − λ2 ) α3 (µ3 − λ3 ) B3 = . α3 (µ23 − λ23 ) α1 (µ21 − λ21 ) − α2 (µ22 − λ22 ) 

α1 (µ21

α3 (µ23 + λ23 ) + λ21 ) − α2 (µ22 + λ22 )

A singular quadric of the net is such that det(M ) = det(B1 ) det(B2 ) det(B3 ) = 0. One eaely check that det(B1 ) = det(B2 ) + det(B3 ). So, if α1 , α2 , α3 are such that det(Bi ) = det(Bj ) = 0 i 6= j, then also for the third block Bh , h 6= i, h 6= j one has det(Bh ) = 0. Hence such a choice corresponds to a quadric of rank 3. There are exactly four possible choices of (α1 , α2 , α3 ) ∈ C3 which satisfy the condition det(Bi ) = 0 for i = 1, 2, 3. Putting λi = 1, i = 1, 2, 3 and p p p w2 = (µ21 − µ23 )(µ21 µ23 − 1), w3 = (µ22 − µ21 )(µ21 µ22 − 1) w1 = (µ22 − µ23 )(µ22 µ23 − 1), the rank 3 quadrics Si correspond to the following choices of (α1 , α2 , α3 ) ∈ C3 : S1 S3

to to

(α1 , α2 , α3 ) = (w1 , w2 , w3 ) (α1 , α2 , α3 ) = (w1 , −w2 , w3 )

S2 S4

to to

(α1 , α2 , α3 ) = (w1 , w2 , −w3 ) (α1 , α2 , α3 ) = (w1 , −w2 , −w3 )

Since for these choices det(Bi ) = 0 for i = 1, 2, 3, the quadrics S1 , S2 , S3 , S4 are of type (β1 x1 + β2 x2 )2 + (β3 x3 + β4 x4 )2 + (β5 x5 + β6 x6 )2 = 0, the singular locus of such a quadric is the plane of P5 :   β1 x 1 + β2 x 2 = 0 β3 x 3 + β4 x 4 = 0  β5 x5 + β6 x6 = 0.

We observe that the singular planes of S1 and S2 are complementary planes in P5 and the same is true for the singular planes of S3 and S4 . Then, up to a change of coordinates, we can assume that: S1 = y12 + y22 + y32 , S2 = z12 + z22 + z32 , S3 = (l1 y1 + m1 z1 )2 + (l2 y2 + m2 z2 )2 + (l3 y3 + m3 z3 )2 A/ι = S1 ∩ S2 ∩ S3 .

The intersection between Sing(S1 ) and S2 is a conic C2 . The intersection of this conic with the hypersurface S3 is made up of four points. So Sing(S1 ) ∩ (A/ι) = Sing(S1 ) ∩ (S1 ∩ S2 ∩ S3 ) = Sing(S1 ) ∩ S2 ∩ S3 is made up of four points which

18


must be singular on A/ι (as A/ι is the complete intersection between S1 , S2 and S3 and the points are in Sing(S1 )). These four points are four nodes of the surface A/ι. There is a complete symmetry between the four quadrics S1 , S2 , S3 , S4 , so we have: Lemma 5.3. On each plane Sing(Si ) there are exactly four singular points of the surface A/ι. Let us now consider the classes of Remark 2.10 described by the set J8 ⊂ (Z/2Z)4 . We call any of them uJ8 . These classes have self intersection −2 and they are effective. Since uJ8 · H = 4, they correspond to rational quartics on A/ι passing through eight nodes of the Psurface. Moreover, they correspond to curves with multiplicity 2, indeed 2uJ8 + ∈J8 Kp is linearly equivalent to H, which is the class of the hyperplane section. The classes of these rational curves and the classes in the Kummer lattice generate the Néron–Severi group of Km(A). These curves are in a certain sense the analogue of the tropes of Km(J(C)): like the tropes of Km(J(C)) they are rational curves obtained as special hyperplane sections of Km(A) and they generate the Néron–Severi group of the Kummer surface together with the curves of the Kummer lattice. The polarization H − Kp1 − Kp2 − Kp3 . Let us choose three singular points pi , i = 1, 2, 3 such that p1 , p2 are contained in Sing(S1 ) and p3 ∈ / Sing(S1 ). These three points generate a plane in P5 . The projection of φH (Km(A)) from this planes is associated to the linear system H − Kp1 − Kp2 − Kp3 . The map φH−Kp1 −Kp2 −Kp3 : Km(A) → P2 is a 2 : 1 cover of P2 ramified along the union of two conics and two lines. The lines are the images of two of the rational curves with classes of type uJ8 , where J8 contains p1 , p2 , p3 ∈ J8 . This description of Km(A) was presented in [G1]. Deformation. This model exhibits Km(A) as a special member of the 6dimensional family of K3 surfaces which are double cover of P2 branched along two conics and two lines. The covering involution is a non-symplectic involution fixing four rational curves. By Nikulin’s classification of non-symplectic involutions (see e.g. [AN, Section 2.3]) it turns out that the generic member of this family of K3 surfaces has Néron–Severi group isometric to h2i ⊕ A1 ⊕ D4⊕3 and transcendental lattice U (2)⊕2 ⊕ h−2i⊕4 (this family is studied in details in [KSTT]). The transcendental lattice U (2)⊕2 ⊕ h−8i of Km(A) clearly embeds in the previous lattice. P The polarization 2H − 21 p∈(Z/2Z)4 Kp . We call this divisor D. It is ample by Proposition 4.3. The projective model φD (Km(A)) is a smooth K3 surface in P13 . The curves of the Kummer lattice and the ones associated to classes of type uJ8 are sent to lines and hence the Néron–Severi group of φD (Km(A)) is generated by lines (cf. Proposition 5.1). P The nef class 12 (H − p∈J4 Kp ). We call it F . By Proposition 4.6, it defines a map φF : Km(A) → P1 which exhibits Km(A) as elliptic fibration with 12 fibers of type I2 and Mordell-Weil group isomorphic to Z3 ⊕ (Z/2Z)2 . Indeed the zero section and three independent sections of infinite order are the curves Ka,b,c,d such that F · Ka,b,c,d = 1. The non trivial components of the 12 fibers P of type I2 are Ke,f,g,h , such that F · Ke,f,g,h = 0. The curves F + 2K0,0,0,0 + ( p∈W3 Kp )/2 and P F + 2K0,0,0,0 + ( p∈W4 Kp )/2 are two 2-torsion sections. This description of an elliptic fibration on Km(A) follows immediately by the properties of the divisors


19

of the Néron–Severi group. However a geometrical construction giving the same result is obtained considering the projection of the model of φH (Km(A)) ⊂ P5 from the plane Sing(S1 ). The image of this projection lies in the complementary plane Sing(S2 ) and is a conic C. Let p be a point of C and let P3p be the space generated by Sing(S1 ) and by p. The fiber over p is S2 ∩ S3 ∩ P3p . The fiber over a generic point of C is an elliptic curve (the intersection of two quadric in P3 ). There are 12 points in C, corresponding to the 12 singular points of φH (Km(A)) which are not on the plane Sing(S1 ), such that the fibers over these points are singular and in fact of type I2 . A geometrical description of this elliptic fibration is provided also in [Me], where it is obtained as double cover of an elliptic fibration on Km(J(C)). Shioda–Inose structure. We now describe the 3-dimensional family of K3 surfaces which admit a Shioda–Inose structure associated to Km(A) as described in Theorem 2.18. It is obtained using results of [vGS, Section 4.6]: consider the K3 surface X with ρ(X) = 17 and admitting an elliptic fibration with fibers I16 + 8I1 and Mordell-Weil group isometric to Z/2Z. By [vGS, Proposition 4.7] the discriminant of N S(X) is 4 and the translation t by the 2-torsion section is a Morrison–Nikulin involution. Thus, the desingularization of X/t is a Kummer surface, which is in fact Km(A) by Theorem 2.18. The elliptic fibration induced on Km(A) has I8 + 8I2 singular fibers and Mordell–Weil group (Z/2Z)2 . Using the curves contained in the elliptic fibration one can easily identify the sublattice N ⊕ E8 (−1) of N S(Km(A)): the lattice N contains the 8 non trivial components of the 8 fibers of type I2 and the lattice E8 (−1) is generated by 7 components of the fiber of type I8 and by the zero section. As in the case of the Jacobian of a curve of genus 2, we give a Z-basis of the Néron– Severi group of Km(A) related to the Shioda–Inose structure and we identify the class of the fiber of this fibration: with the P previous notation a Z-basis is given by {h(Q+N1 +N2 +N3 +N4 i)/2, N1 , . . . , N7 , 8 Ni /2, E1 , . . . , E8 }, where Q2 = 8 and i=1 Q is orthogonal to N ⊕E8 (−1); the class of the fiber in terms of the previous basis of the Néron–Severi group is F := Q−5E1 −10E2 −15E3 −12E4 −9E5 −6E6 −3E7 −8E8 . 5.3. Kummer surface of a (1, 3) polarized Abelian surface. Let A be a (1, 3) polarized Abelian surface, then N S(A) = ZL, L2 = 6. The polarization H. The model of the singular quotient A/ι is associated to the divisor H in N S(Km(A)) with H 2 = 12. By Proposition 4.1 and [SD, Theorem 5.2] this model is a singular K3 surface in P7 . Let us now consider the 16 classes of Remark 2.10 associated to the set J10 ⊂ (Z/2Z)4 . We call any of them uJ10 . They are (−2)-classes (see Remark 2.10) and are sent to rational curves of degree 6 on φH (Km(A)). P The polarization H − 12 ( p∈(Z/2Z)4 Kp ). We call it D. It is ample by Proposition 4.3 and since D2 = 4, the surface φD (Km(A)) is a smooth quartic in P3 . The curves of the Kummer lattice and the curves associated to uJ10 are sent to lines. Since the classes of the curves in the Kummer lattice and the classes uJ10 generate the Néron–Severi group of Km(A), the Néron–Severi group of φD (Km(A)) is generated by lines (cf. Proposition 5.1). The polarization H −K0,0,1,0 −K0,0,1,1 −K1,0,0,0 −K0,1,0,0 −K0,0,1,1 . It defines a 2 : 1 map from Km(A) to P2 , since 11 curves Kp are contracted the branch locus is a reducible sextic with 11 nodes.

20


Deformation. The generic K3 surface double cover of P2 branched on a reducible sextic with 11 nodes lies in a 8-dimensional family and has transcendental lattice equal to U (2)⊕2 ⊕ h−2i⊕6 , see [AN, Section 2.3]. Clearly the transcendental lattice U (2)⊕2 ⊕ h−12i can be primitively embedded in U (2)⊕2 ⊕ h−2i⊕6 , so the family of Kummer surfaces of a (1, 3)-polarized Abelian surface is a special 3-dimensional subfamily. P The nef class 21 (H − p∈J6 Kp ). We call it F . By Proposition 4.6 it defines an elliptic fibration Km(A) → P1 with 10 fibers of type I2 : the components of these fibers not meeting the zero section are the curves Ka,b,c,d of the Kummer lattice such that F · Ka,b,c,d = 0. The Mordell–Weil group is Z5 and the curves Ke,f,g,h such that F · Ke,f,g,h = 1 are the zero section and 5 sections of infinite order (but they are not the Z-generators of the Mordell–Weil group). Shioda–Inose structure.We now describe the 3-dimensional family of K3 surfaces which admit a Shioda–Inose structure associated to Km(A) as described in Theorem 2.18. It was already described independently in [G1, Remark 3.3.1 (Section 3.3)] and [Koi, Section 3.1]. Let us consider the K3 surfaces X with ρ(X) = 17 and with an elliptic fibration with reducible fibers I6∗ + I6 and Mordell–Weil group Z/2Z (as in the arXiv version of the paper [Shim, Table 1, nr. 1357]). The translation t by the 2-torsion section is a Morrison–Nikulin involution (in fact it is immediate to check that it switches two orthogonal copies of E8 (−1) ⊂ N S(X)) and hence the desingularization of the quotient X/t is a Kummer surface. The latter admits an elliptic fibration induced by the one on X, with reducible fibers I3∗ + I3 + 6I2 and a 2-torsion section. By the Shioda-Tate formula (see e.g. [Sh1, Corollary 1.7]) the discriminant of the Néron–Severi group of such an elliptic fibration is (4 · 3 · 26 )/22 and thus this Kummer surface is the Kummer surface of a (1, 3)-polarized Abelian surface. As in the case of the Jacobian of a curve of genus 2, we give a Z-basis of the Néron–Severi group of Km(A) related to the Shioda–Inose structure and we can identify the class of the fiber of this fibration: the 8 curves Ni are the 6 non trivial components of each fiber of type I2 and 2 non trivial components of I3∗ with multiplicity 1; the curves Ei are the zero section, two components of I3 and five components of I3∗P . With the previous notation a Z-basis is given by {h(Q + N1 + N2 i)/2, N1 , . . . N7 , 8 Ni /2, E1 , . . . , E8 }, where Q2 = 12 i=1 and Q is orthogonal to N ⊕E8 (−1); the class of the fiber in terms of this basis of the Néron–Severi group is F := Q−6E1 −12E2 −18E3 −15E4 −12E5 −8E6 −4E7 −9E8 . 6. K3 surfaces with symplectic action of the group (Z/2Z)4 and their quotients In the following sections we study two 4-dimensional families of K3 surfaces that contain subfamilies of Kummer surfaces. Indeed, we have seen that every Kummer surface admits a symplectic action of the group (Z/2Z)4 (Proposition 3.3), but the moduli space of K3 surfaces with symplectic action by (Z/2Z)4 has dimension 4 and thus the Kummer surfaces are a 3-dimensional subfamily. We will also study the family of K3 surfaces obtained as desingularization of the quotient of a K3 surface by the group (Z/2Z)4 acting symplectically on it. By Proposition 3.3 this family also contains the 3-dimensional family of Kummer surfaces. Let G = (Z/2Z)4 be a group of symplectic automorphisms on a K3 surface X. We observe that G contains (24 − 1) = 15 symplectic involutions so we have


21

8 · 15 = 120 distinct points with non trivial stabilizer group on X, and these are all the points with a non trivial stabilizer on X (cf. [Ni3, Section 5]). Moreover we have a commutative diagram: β e −→ X X π↓ ↓ π′ β˜ Y −→ Y¯ ,

(3)

e is the blow up of X at the 120 points where Y¯ is the quotient of X by G, X with non trivial stabilizer (hence it contains 120 (−1)-curves) and Y is the minimal e by the induced resolution of the quotient Y¯ and simultaneously the quotient of X action. Observe that Y contains 15 (−2)-curves coming from the resolution of the singularities. In fact each fixed point on X has a G-orbit of length 8. In particular the rank of the Néron–Severi group of Y is at least 15 and in fact 16 if X, and so Y , is algebraic. In particular, since by [I, Corollary 1.2] rank N S(X)=rank N S(Y ), a K3 surface with a symplectic action of (Z/2Z)4 has at least Picard number 15 (16 if it is algebraic). Finally π is 16 : 1 outside the branch locus. 7. K3 surfaces with symplectic action of (Z/2Z)4 In this section we analyze the K3 surface X admitting a symplectic action of (Z/2Z)4 , in particular we identify the possible Néron–Severi groups of such a K3 surface if the Picard number is 16, which is the minimum possible for an algebraic K3 surface with this property. This allows us to describe the families of such K3 surfaces (cf. Corollary 7.11) and to prove that every K3 surfaces admitting (Z/2Z)4 as group of symplectic automorphisms also admits an Enriques involution: this generalizes the similar result for Kummer surfaces given in Proposition 3.1. 7.1. The N´ eron–Severi group of X. Theorem 7.1. (cf. [G1]) Let X be an algebraic K3 surface with a symplectic ⊕3 action of (Z/2Z)4 and let Ω⊥ be the invariant lattice (Z/2Z)4 =< −8 > ⊕U (2) 4

H 2 (X, Z)(Z/2Z) . Then ρ(X) ≥ 16. If ρ(X) = 16, denote by L a generator of (Ω(Z/2Z)4 )⊥ ∩ N S(X) with L2 = 2d > 0. Let L2d (Z/2Z)4 := ZL ⊕ Ω(Z/2Z)4 ⊂ N S(X). 2d Denote by L′2d (Z/2Z)4 ,r an overlattice of L(Z/2Z)4 of index r. Then there are the following possibilities for d, r and L. 1) If d ≡ 0 mod 2 and d 6≡ 4 mod 8, then r = 2, L = w1 := (0, 1, t, 0, 0, 0, 0) ∈ 2 2 Ω⊥ (Z/2Z)4 and L = w1 = 4t.

2) If d ≡ 4 mod 8 and d 6≡ −4 mod 32, then: 2 2 either r = 2, L = w1 := (0, 1, t, 0, 0, 0, 0) ∈ Ω⊥ (Z/2Z)4 and L = w1 = 4t, 2 2 or r = 4, L = w2 := (1, 2, 2s, 0, 0, 0, 0) ∈ Ω⊥ (Z/2Z)4 and L = w2 = 8(2s − 1). 3) If d ≡ −4 mod 32 then: 2 2 either r = 2, L = w1 := (0, 1, t, 0, 0, 0, 0) ∈ Ω⊥ (Z/2Z)4 and L = w1 = 4t, ⊥ 2 2 or r = 4, L = w2 := (1, 2, 2s, 0, 0, 0, 0) ∈ Ω(Z/2Z)4 and L = w2 = 8(2s − 1), 2 2 or r = 8, L = w3 := (1, 4, 4u, 0, 0, 0, 0) ∈ Ω⊥ (Z/2Z)4 and L = w3 = 8(8u−1).

22


If N S(X) is an overlattice of Zw1 ⊕ Ω(Z/2Z)4 , then TX ≃ h−8i ⊕ h−4ti⊕ U (2)⊕2 ; −8 4 ⊕ U (2)⊕2 ; If N S(X) is an overlattice of Zw2 ⊕ Ω(Z/2Z)4 , then TX ≃ 4 −4s −8 2 If N S(X) is an overlattice of Zw3 ⊕ Ω(Z/2Z)4 , then TX ≃ ⊕ U (2)⊕2 . 2 −4u Proof. Since Ω(Z/2Z)4 ⊂ N S(X) and X is algebraic we have ρ(X) ≥ 16. The proof of the unicity of the possible overlattices of L2d (Z/2Z)4 is based on the following idea. Let us consider the lattice orthogonal to Ω(Z/2Z)4 in ΛK3 . For each element ⊥ s(= L) ∈ Ω⊥ (Z/2Z)4 in a different orbit under isometries of Ω(Z/2Z)4 , we can consider the lattice Zs ⊕ Ω(Z/2Z)4 . To compute the index of the the overlattice R(= N S(X)) of Zs ⊕ Ω(Z/2Z)4 which is primitively embedded in ΛK3 , we consider the lattice ⊥ R⊥ = s⊥ ∩ Ω⊥ ⊂ ΛK3 (which is isometric to TX ). We (Z/2Z)4 = (Zs ⊕ Ω(Z/2Z)4 ) ) ⊥ compute then the discriminant group of R to get the discriminant group of R and so we get the index r of Zs ⊕ Ω(Z/2Z)4 in R(= N S(X)). Recall that 3 3 Ω⊥ (Z/2Z)4 ≃ h−8i ⊕ U (2) ≃ (h−4i ⊕ U )(2).

The orbits of elements by isometries of this lattice are determined by the orbits of elements by isometries of the lattice h−4i ⊕ U 3 . In the next sections we investigate them, then the proof of the theorem follows from the results of Section 7.2. We remark moreover that under our assumptions two overlattices Ri ⊃ Zwi ⊕ Ω(Z/2Z)4 and Rj ⊃ Zwj ⊕ Ω(Z/2Z)4 , i 6= j, cannot be isometric in ΛK3 since their orthogonal complements Ri⊥ and Rj⊥ are different. These are determined in Proposition 7.8 below and they are the transcendental lattices TX in our statement. 7.2. The lattice h−2di ⊕ U ⊕ U . Lemma 7.2. Let (a1 , a2 , a3 , a4 ) be a vector in the lattice U ⊕ U . There exists an isometry which sends the vector (a1 , a2 , a3 , a4 ) to the vector (d, de, 0, 0). In particular the vector (a1 , a2 , 0, 0) can be sent to (d, de, 0, 0) where d = gcd(a1 , a2 ) and d2 e = a1 a2 . Proof. The lattice U ⊕ U is isometric to the lattice {M (2, Z), 2 det} of the square matrices of dimension two with bilinear form induced by the quadratic form given by the determinant multiplied by 2. Explicitly the isometry can be written as a1 a3 a1 −a3 U ⊕ U −→ M (2, Z), , 7→ . a2 a4 a4 a2 It is well known that under the action of the orthogonal group O(M (2, Z)) each matrix of M (2, Z) can be sent in a diagonal matrix with diagonal (d1 , d2 ), d1 |d2 (this is the Smith Normal Form). Thus the lemma follows. Lemma 7.3. There exists an isometry which sends the primitive vector (a0 , a1 , a2 , a3 , a4 ) ∈ T2d := h−2di ⊕ U ⊕ U , to a primitive vector (a, d, de, 0, 0) ∈ h−2di ⊕ U ⊕ U . Proof. The primitive vector (a0 , a1 , a2 , a3 , a4 ) is sent to a primitive vector by any isometry. By Lemma 7.2 there exists an isometry sending (a1 , a2 , a3 , a4 ) ∈ U ⊕ U to (d, de, 0, 0) ∈ U ⊕ U , thus there exists an isometry sending (a0 , a1 , a2 , a3 , a4 ) to (a0 , d, de, 0, 0) and (a0 , d, de, 0, 0) is primitive. The previous lemma allows us to restrict our attention to the vectors in the lattice A2d := h−2di ⊕ U .


23

Lemma 7.4. There exists an isometry of A2d which sends the vector (a, 1, c), to the vector (0, 1, r), where 2c − 2da2 = 2r. Proof. First we observe that (a, 1, c) · (a, 1, c) = (0, 1, r) · (0, 1, r) = 2r. Let Rv denote the reflection with respect to v = (1, 0, d), then for w = (x, y, z) we have   −x + y w·v . y Rv (w) = w − 2 v= v·v −2dx + dy + z

If a > 0 we apply the reflection Rv to (a, 1, c), (v = (1, 0, d)):     1−a a . 1 Rv  1  =  −2da + d + c c

Let D be the isometry of A2d ,

Then



−1 0 D= 0 1 0 0

 0 0 . 1

   a−1 a . 1 D ◦ Rv  1  =  −2da + d + c c Applying a times the isometry D ◦ Rv we obtain     0 a (D ◦ Rv )a  1  =  1  . 2r c 

Lemma 7.5. There exists an isometry of A2d which sends a vector q2 := (wh ± j, w, wt), with t, h ∈ Z, w, j ∈ N, 0 < j ≤ xd/2y to the vector p2 := (j, w, s), where s = −dwh2 ∓ 2dhj + wt. Proof. Without lost of generality we can assume h > 0 (if h ≤ 0, it is sufficient to consider the action of D). Let us apply the isometry D ◦ Rv to the vector q2 :     w(h − 1) ± j wh ± j . = w w (D ◦ Rv )  −2d(wh ± j) + dw + wt wt

As in the previous proof, applying D ◦ Rv decreases the first component and the second remains the same. Applying h-times the isometry to q2 , we obtain that the first component is j or −j. In the second case we apply again the isometry D, and so in both situations we obtain p2 . Lemma 7.6. Let p be a prime number. Let us consider the lattice T2p = h−2pi ⊕ U ⊕ U . There exists an isometry of T2p which sends the vector q := (n, b, bf, 0, 0), b ∈ Z>0 , n ∈ N, gcd(n, b) = 1 in one of the following vectors: • v1 = (0, 1, r, 0, 0) where 2b2 f − 2pn2 = 2r; • v2 = (1, 2, 2s, 0, 0), where 2b2 f − 2pn2 = 8s − 2p; • vp = (l, p, pt, 0, 0), where 2b2 f − 2pn2 = 2pt − 2pl2 , 0 < l ≤ xp/2y; • v2p = (j, 2p, 2pu, 0, 0), where 2b2 f −2pn2 = 8p2 u−2pj 2 , 0 < j < p, j ≡ 1 mod 2.

24


Proof. We can assume n ∈ N and b > 0 (if it is not the case it suffices to consider the action of − id and of D). Let us consider the reflection Rv , associated to the vector v = (1, 0, p, 0, 0). We have     n −n + b  b    b        Rv  bf  =  −2pn + pb + bf  .  0    0 0 0

Again we can change the sign of the first component and we obtain (b−n, b, −2pn+ pb + bf, 0, 0). By Lemma 7.2 this vector can be transformed in (b − n, b1 , b1 f1 , 0, 0), where gcd(b, −2pn + pb + bf ) = b1 . Then b1 ≤ b := b0 . We apply now Lemma 7.2 to the vector (n1 , b1 , b1 f1 , 0, 0) with n1 := |b − n| > 0 (eventually change the sign of bn by using the matrix D). The second component of the vector b1 is a positive number, so after a finite number of transformations there exists η such that bη = bη+1 and gcd(nη , bη ) = 1. Since bη |(pbη + bη f ) and gcd(nη , bη ) = 1 (recall that the image of a primitive vector by an isometry is again primitive) bη = bη+1 if and only if bη divides 2p, i.e. if bη = 1, 2, p, 2p. Moreover gcd(bη −nη , bη ) = 1. So by Lemma 7.3 and applying eventually the transformation D to get the first component of the vector positive, after a finite number of transformations we obtain that q is isometric to one of the vectors (a, 1, f ′ , 0, 0), (2k + 1, 2, 2f ′ , 0, 0), (ph ± l, p, pf ′ , 0, 0), (2pk±j, 2p, 2pf ′ , 0, 0). Applying Lemma 7.4 and 7.5 we obtain that these vectors are isometric respectively to (0, 1, r, 0, 0), (1, 2, 2s, 0, 0), (l, p, pt, 0, 0) (j, 2p, 2pu, 0, 0). Remark 7.7. The vector (ts, t, f, 0, 0) is isometric to (0, t, ∗, 0, 0) by applying stimes Rv ◦ D. Proposition 7.8. Let p be a prime number. The orbits of the following vectors of T2p under isometries of T2p are all disjoint: • v0 = (1, 0, 0, 0, 0); • v1 = (0, 1, r, 0, 0); • v2 = (1, 2, 2s, 0, 0); • vp = (l, p, pt, 0, 0), where 0 < l ≤ xp/2y; • v2p = (j, 2p, 2pu, 0, 0), where 0 < j < p, j ≡ 1 mod 2. Proof. If two vectors x, y of T2p are isometric, then x2 = y 2 and the discriminants of the lattices orthogonal to x and y are equal: d(x⊥ ) = d(y ⊥ ). We resume the properties of the vectors vi in the following table: v v2

v0 −2p

v1 2r

v2 −2p + 8s −2p p ⊥ ⊕U v U ⊕ U h−2pi ⊕ h−2ri ⊕ U p −2s ⊥ d(v ) 1 −4pr −p(4s − p)

vp 2 −2pl + 2p2 t −2p 2l ⊕U 2l −2t 2 −4(pt − l )

v2p 2 −2pj + 8p2 u −2p j ⊕U j −2u 2 −4pu + j

For each copy of vectors x and y chosen from v0 , v1 , v2 , vp , v2p the conditions x2 = y 2 and d(x⊥ ) = d(y ⊥ ) are incompatible. For example let us analyze the case of vp and v2p , the other cases are similar. We have: −2pl2 + 2p2 t = −2pj 2 + 8p2 u and 4(pt − l2 ) = −4pu + j 2 . By the first equation −l2 + pt = −j 2 + 4pu. Substituting in the second equation we obtain 5(pt − l2 ) = 0 and so pt = l2 . This implies p|l2 and so p|l. Since l ≤ xp/2y this is impossible.


25

The previous results imply the following proposition: Proposition 7.9. A primitive vector (a0 , a1 , a2 , a3 , a4 ) of the lattice h−2pi⊕U ⊕U is isometric to exactly one of the vectors: • v0 = (1, 0, 0, 0, 0); • v1 = (0, 1, r, 0, 0) where 2a1 a2 + 2a3 a4 − 2pa20 = 2r; • v2 = (1, 2, 2s, 0, 0) where 2a1 a2 + 2a3 a4 − 2pa20 = −2p + 8s; • vp = (l, p, pt, 0, 0), where 0 < l ≤ xp/2y and 2a1 a2 + 2a3 a4 − 2pa20 = −2pl2 + 2p2 t; • v2p = (j, 2p, 2pu, 0, 0), where 0 < j ≤ p and 2a1 a2 + 2a3 a4 − 2pa20 = −2pj 2 + 8p2 u. Remark 7.10. In particular in the case p = 2 the only possibilities are the vectors (1, 0, 0, 0, 0), (0, 1, r, 0, 0), (1, 2, 2s, 0, 0) and (1, 4, 4u, 0, 0). 7.3. The family. Let us denote by L2d r,wi the overlattice of index r of Zwi ⊕Ω(Z/2Z)4 , with wi2 = 2d described in Theorem 7.1. If X is a K3 surface such that N S(X) ≃ L2d r,wi for a certain r = 2, 4, 8 and i = 1, 2, 3, then Ω(Z/2Z)4 is clearly primitively embedded in N S(X) and thus X admits (Z/2Z)4 as group of symplectic automorphisms (cf. [Ni3, Theorem 4.15]). Hence, the lattices L2d r,wi determine the family of algebraic K3 surfaces admitting a symplectic action of (Z/2Z)4 . More precisely: Corollary 7.11. The families of algebraic K3 surfaces admitting a symplectic action of (Z/2Z)4 are the families of L2d r,wi -polarized K3 surfaces, for a certain r = 2, 4, 8, i = 1, 2, 3, d ∈ 2N>0 . In particular the moduli space has a countable numbers of connected components of dimension 4. Remark 7.12. If one fixes the value of d, then there is a finite number of possibilities for r and wi : for example if d = 2, then r = 2 and i = 1, w1 = (0, 1, 1, 0, 0, 0, 0). This implies that the family of quartic surfaces in P3 admitting a symplectic action of (Z/2Z)4 has only one connected component of dimension 4. In [E] the family of quartics invariant for the Heisenberg group(≃ (Z/2Z)4 ) is described and since it is a 4-dimensional family of K3 surfaces admitting (Z/2Z)4 as group of symplectic automorphisms we conclude that the family presented in [E] is the family of the (L42,w1 )-polarized K3 surfaces. The Néron–Severi group of such a K3 surfaces are generated by conics as proved in [E, Corollary 7.4]. 7.4. The subfamily of Kummer surfaces. By Corollary 2.9, for every non negative integer d there exists a connected component of the moduli space of Kummer ′ surfaces, which we called Fd and is the family of the K4d -polarized K3 surfaces. For every d the component Fd is 3-dimensional and by Proposition 3.3 it is contained in a connected component of the moduli space of K3 surfaces X admitting G as group of symplectic automorphisms. The following proposition identifies the components of the moduli space of K3 surfaces with a symplectic action of G which contain Fd : ′ Proposition 7.13. The family of the K4d -polarized Kummer surfaces is a codi 4d mension one subfamily of the following families: the L2,w1 -polarized K3 surfaces; 8(2d−1) 8(8d−1) the L4,w2 -polarized K3 surfaces; the L8,w3 -polarized K3 surfaces. ′ or Proof. It suffices to show that there exists a primitive embedding Lhi,wj ⊂ K4d ⊥ ′ ⊥ equivalently a primitive embedding (K4d ) ⊂ Lhi,wj for (i, j, h) = (2, 1, 4d), (4, 2, 8(2d− ⊥ ′ ⊥ ) ≃ h−4di ⊕ U (2) ⊕ U (2) and Lhi,wj 1)), (8, 3, 8(8d − 1)). We recall that (K4d

26


is the transcendental lattice of the generic K3 surface X described in Theorem 7.1. With the notation of Theorem 7.1 sending a basis of h−4di⊕U (2)⊕U (2) to the basis vectors (0, 1, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1) ⊥ with t = d, s = d, u = d if i = 2, 4, 8 respectively, we obtain an explicit of Lhi,wj ⊥ ′ ⊥ primitive embedding of (K4d ) in Lhi,wj .

We observe that the sublattice of N S(Km(A)) invariant for the action induced by the translation by the two torsion points on A, i.e.,P invariant for the action of G defined in Proposition 3.3, is generated by H and 12 ( p∈(Z/2Z)4 Kp ). Indeed H is the image of the generator of N S(A) by the map πA∗ P, with the notation of diagram (1). Thus, the lattice Ω(Z/2Z)4 is isometric to hH, 12 ( p∈(Z/2Z)4 Kp )i⊥ ∩N S(Km(A)) and in fact the lattice L2d 2,w1 (which contains Ω(Z/2Z)4 and an ample class) is isometric P to h 21 ( p∈(Z/2Z)4 Kp )i⊥ ∩ N S(Km(A)). Remark 7.14. The previous proposition implies that the family of the Kummer surfaces of a (1, d)-polarized Abelian surface is contained in at least three distinct connected components of the family of K3 surfaces admitting a symplectic action of G. In particular, the intersection among the connected components of such family of K3 surfaces is non empty and of dimension 3.

7.5. Enriques involution. In Section 3 we have seen the result of Keum, [Ke2]: Every Kummer surface admits an Enriques involution. We now prove that this property holds more in general for the K3 surfaces admitting (Z/2Z)4 as group of symplectic automorphisms and minimal Picard number. Theorem 7.15. Let X be a K3 surface admitting (Z/2Z)4 as group of symplectic automorphisms and such that ρ(X) = 16, then X admits an Enriques involution. Proof. By Proposition 3.2 it suffices to prove that the transcendental lattice of X admits a primitive embedding in U ⊕ U (2) ⊕ E8 (−2) whose orthogonal does not contain vectors of length −2. The existence of this embedding can be proved as in [Ke2]. We Q be one of the following lattices: the proof. Let briefly sketch −4 1 −4 2 . The transcendental lattice of X is , h−4i ⊕ h−2ti, 1 −2u 2 −2s (U 2 ⊕ Q)(2). It suffices to prove that there exists a primitive embedding of U (2) ⊕ Q(2) in U ⊕ E8 (−2). The lattice h−2i ⊕ Q is an even lattice with signature (0, 3). By [Ni2, Theorem 14.4], there exists a primitive embedding of h−2i ⊕ Q in E8 (−1), which induces a primitive embedding of h−4i ⊕ Q(2) in E8 (−2). Let b1 , b2 , b3 be the basis of h−4i ⊕ Q(2) in E8 (−2). Let e and f be a standard basis of U (i.e. e2 = f 2 = 0, ef = 1). Then the vectors e, e + 2f + b1 , b2 , b3 give a primitive embedding of U (2) ⊕ Q(2) in U ⊕ E8 (−2) whose orthogonal complement does not contain vectors of length −2 (cf. [Ke2, §2, Proof of Theorem 2]). 8. The quotient K3 surface The surface Y obtained as desingularization of the quotient X/(Z/2Z)4 contains 15 rational curves Mi , which are the resolution of the 15 singular points of type A1 on X/(Z/2Z)4 . The minimal primitive sublattice of N S(Y ) containing these curves is denoted by M(Z/2Z)4 . It is described in [Ni3, Section 7] as an overlattice of the lattice hMi ii=1,...,15 of index 24 .


27

Proposition 8.1. Let Y be a K3 surface such that there exists a projective K3 ^ 4 . Then surface X and a symplectic action of (Z/2Z)4 on X with Y = X/(Z/2Z) ρ(Y ) ≥ 16. ⊥N S(Y ) Moreover if ρ(Y ) = 16, let L = M(Z/2Z) 4 . Then N S(Y ) is an overlattice of index 2 2 of ZL ⊕ M(Z/2Z)4 , where L = 2d > 0. In particular, N S(Y ) is generated by ∨ ZL ⊕ M(Z/2Z)4 and by a class (L/2, v/2), v/2 ∈ M(Z/2Z) 4 /M(Z/2Z)4 (that is not ∨ 2 2 trivial in M(Z/2Z)4 /M(Z/2Z)4 ), L ≡ −v mod 8. Proof. A K3 surface Y obtained as desingularization of the quotient of a K3 surface X by the symplectic group of automorphisms (Z/2Z)4 , has M(Z/2Z)4 ⊂ N S(Y ). Since M(Z/2Z)4 is negative definite and Y is projective (it is the quotient of X, which is projective), there is at least one class in N S(Y ) which is not in M(Z/2Z)4 so ρ(Y ) ≥ 1 + rank M(Z/2Z)4 = 16. In particular if ρ(Y ) = 16, then the orthogonal complement of M(Z/2Z)4 in N S(Y ) is generated by a class with a positive self intersection, hence N S(Y ) is either ZL ⊕ M(Z/2Z)4 or an overlattice of ZL ⊕ M(Z/2Z)4 with a finite index. The discriminant group of M(Z/2Z)4 is (Z/2Z)7 by [Ni3, Section 7] and so the discriminant group of the lattice ZL ⊕ M(Z/2Z)4 is (Z/2dZ) ⊕ (Z/2Z)7 . It has eight generators. If the lattice ZL ⊕ M(Z/2Z)4 is the Néron–Severi group of a K3 surface Y , then also the discriminant group of TY has eight generators, but TY has rank 22 − ρ(Y ) = 6, so this is impossible. Hence N S(Y ) is an overlattice of ZL ⊕ M(Z/2Z)4 . The index of the inclusion and the costruction of the overlattice can be computed as in [GSa1, Proposition 2.1] or as in Theorem 2.7. The Kummer surfaces are also examples of K3 surfaces obtained as desingularization of the quotient of K3 surfaces by the action of (Z/2Z)4 as group of symplectic automorphisms, see Proposition 3.3. In [G2, Sections 4.2, 4.3] the action of G on the Kummer lattice and the con^ are described. The images of the curves Ka,b,c,d , struction of the surface Km(A)/G 4 (a, b, c, d) ∈ (Z/2Z) on Km(A) under the quotient map Km(A) −→ Km(A)/G is a single curve. This curve can be naturally identified with the curve K0,0,0,0 ^ ∼ on the minimal resolution Km(A)/G = Km(A) (see [G2]). The minimal resolution contains also fifteen (−2)-curves coming from the blowing up of the nodes on Km(A)/G, which can be identified with Ke,f,g,h , (e, f, g, h) ∈ (Z/2Z)4 \{(0, 0, 0, 0)}. ⊥ These are the fifteen (−2)-curves in M(Z/2Z)4 , hence M(Z/2Z)4 = K(0,0,0,0) ∩ K. This identification allows us to identify the curves of M(Z/2Z)4 with the points of the space (Z/2Z)4 \{(0, 0, 0, 0)}, hence we denote them by Ma,b,c,d , (a, b, c, d) ∈ (Z/2Z)4 \{(0, 0, 0, 0)}. More explicitly, we are identifying the curve Ka,b,c,d with the curve Ma,b,c,d for any (a, b, c, d) ∈ (Z/2Z)4 \{(0, 0, 0, 0)}. By [Ni3] the lattice M(Z/2Z)4 contains the 15 curves Ma,b,c,d , (a, b, c, d) ∈ (Z/2Z)4 \{(0, 0, 0, 0)}, it is generated by 11 of these curves and by 4 other classes which are linear combination of these curves with rational coefficients. These 4 classes have to be contained also in K (because M(Z/2Z)4 ⊂ K) and hence they correspond to hyperplanes in (Z/2Z)4 which do not contains the point (0, 0, 0, 0) (because K(0,0,0,0) 6∈ M(Z/2Z)4 ). P 1 ¯ W (resp. M ¯ W ) denotes 1 P From now on K p∈W Kp (resp. 2 p∈W Mp ) for a sub2 set W of (Z/2Z)4 (resp. W a subset of (Z/2Z)4 \{(0, 0, 0, 0)}). We determine the orbits of elements in the discriminant group of M(Z/2Z)4 and its isometries using the ones of K.

28


Proposition 8.2. With respect to the group of isometries of M(Z/2Z)4 there are ∨ exactly six distinct orbits in the discriminant group M(Z/2Z) 4 /M(Z/2Z)4 . Proof. Let W be one of the following subspaces: 1) W = (Z/2Z)4 ; 2) W is a hyperplane in (Z/2Z)4 ; 3) W is a 2-dimensional plane in (Z/2Z)4 ; 4) W = V ∗ V ′ where V and V ′ are 2-dimensional planes and V ∩ V ′ is a point. ¯ W are in K ∨ and if W is as in 1) or 2) the classes By Remark 2.3 the classes K ¯ KW ∈ K, and thus they are trivial in K ∨ /K. If W is such that (0, 0, 0, 0) ∈ / W , then ¯W = K ¯ W is contained in M ∨ the class M . Indeed it is a linear combination with 4 (Z/2Z) rational coefficients of the curves M(a,b,c,d) with (a, b, c, d) ∈ (Z/2Z)4 \{(0, 0, 0, 0)}, i.e. it is in M(Z/2Z)4 ⊗ Q. Moreover it has an integer intersection with all the classes in K and so in particular with all the classes in M(Z/2Z)4 ⊂ K, i.e. it is in ∨ M(Z/2Z) 4 . We observe that if W is a hyperplane (as in case 2)) and it is such that ¯ W is a class in M(Z/2Z)4 (and hence trivial in the (0, 0, 0, 0) ∈ / W , then the class M discriminant group, see Remark 2.3). ¯ W ′ is a class in If (0, 0, 0, 0) ∈ W , let W ′ be W ′ = W − {(0, 0, 0, 0)}. The class M ∨ ¯ M(Z/2Z)4 . Indeed it is clear that MW ′ ∈ M(Z/2Z)4 ⊗ Q has an integer intersection with all the classes M(a,b,c,d) ∈ M(Z/2Z)4 , (a, b, c, d) ∈ (Z/2Z)4 \{(0, 0, 0, 0)}. Let Z ¯ Z ∈ M(Z/2Z)4 be a hyperplane of (Z/2Z)4 which does not contain (0, 0, 0, 0). Since M ∨ ¯ ¯ ¯ we have to check that MW ′ · MZ ∈ Z. We recall that KW is in K and so it has ¯ Z . This means that W ∩ Z is made up an integer intersection with all the classes K of an even number of points. Since (0, 0, 0, 0) ∈ / Z, (0, 0, 0, 0) ∈ / W ∩ Z and hence ¯W′ · M ¯ Z ∈ Z. W ′ ∩ Z is an even number of points. This implies that M ∨ ∨ ¯ ¯ ¯ If MW ∈ M(Z/2Z)4 , hence either KW or KW ∪{(0,0,0,0)} is in K . Indeed by Remark 2.3 the Kummer lattice is generated by the curves K(a,b,c,d) , (a, b, c, d) ∈ (Z/2Z)4 , ¯ W where Wi is the hyperplane ai = 0, i = 0, 1, 2, 3 (see the by 4 classes of type K i ¯ (Z/2Z)4 . This is clearly equivalent to notation of Remark 2.3) and by the class K ¯ W ′ where say that K is generated by the curves K(a,b,c,d) , by 4 classes of type K i ′ ∨ ¯ ¯ Wi is the hyperplane ai = 1 and by the class K(Z/2Z)4 . If MW ∈ M(Z/2Z)4 , then ¯W · M ¯W′ = K ¯W · K ¯ W ′ ∈ Z. Moreover, since (0, 0, 0, 0) ∈ M / Wi′ , we have also i i ¯ W ∪{(0,0,0,0)} · K ¯ W ′ ∈ Z. To conclude that either K ¯ W or K ¯ W ∪{(0,0,0,0)} is in K ∨ , K i ¯ ¯ (Z/2Z)4 ∈ Z. ¯ ¯ it suffices to prove either that KW · K(Z/2Z)4 ∈ Z or KW ∪{(0,0,0,0)} · K ¯ ¯ This is clear, indeed KW · K(Z/2Z)4 ∈ Z if and only if W consists of an even number of points. If it is not, clearly W ∪ {(0, 0, 0, 0)} consists of an even number of points. ¯ W are in M ∨ Thus, the classes M (Z/2Z)4 for the following subspaces: W = (Z/2Z)4 \{(0, 0, 0, 0)}; W is an hyperplane in (Z/2Z)4 , (0, 0, 0, 0) ∈ / W; W \{(0, 0, 0, 0)} where W is a hyperplane in (Z/2Z)4 , (0, 0, 0, 0) ∈ W ; W is a 2-dimensional plane in (Z/2Z)4 and (0, 0, 0, 0) ∈ / W; W \{(0, 0, 0, 0)} where W is a 2-dimensional plane in (Z/2Z)4 and (0, 0, 0, 0) ∈ W; 4a) W = V ∗ V ′ where V and V ′ are 2-dimensional planes and V ∩ V ′ is a point, (0, 0, 0, 0) ∈ / V ∗ V ′; 4b) W \{(0, 0, 0, 0)} where W = V ∗ V ′ , V and V ′ are 2-dimensional planes and V ∩ V ′ is a point, (0, 0, 0, 0) ∈ V ∗ V ′ .

1) 2a) 2b) 3a) 3b)


29

∨ Each of these cases corresponds to a class of equivalence in the quotient M(Z/2Z) 4 /M(Z/2Z)4 , here we consider these equivalence classes. We will denote with H an hyperplane of ¯ W ∗H ≡ M ¯W + M ¯ H mod ⊕p (Z/2Z)4 such that (0, 0, 0, 0) ∈ / H. We observe that M ¯ W and M ¯ W ∗H coincide in M ∨ ZMp . Clearly the two classes M (Z/2Z)4 /M(Z/2Z)4 if ¯ MH ∈ M(Z/2Z)4 . Let n be the cardinality of W ∩ H, m be the number of curves ¯ W ∗H with a rational, non integer coefficient. In the folM(a,b,c,d) appearing in M ∨ lowing table we resume the classes of M(Z/2Z) 4 which coincide modulo M(Z/2Z)4 and for each of them we give the value discr of the discriminant form on it. The first ¯ W and we put a 0 for n; value of m in the table is the number of curves in M

Case 1); 2b) 3a) 3b) 4a) 4b)

n 0; 0, 8; 3 0, 4, 2, 0 0, 0, 2 4, 2 4, 2

m 15; 7, 7; 7 4, 4, 8, 12 3, 7, 11 6, 10 5, 9

discr 1 2

0 1 2

1 − 21

Indeed by Remark 2.3 the orbit of elements in the discriminant group AK of K are three up to isometries. To prove the latter one considers the action of the group GL(4, Z/2Z) on (Z/2Z)4 , which in fact we can identify with a subgroup of O(AK ). Since GL(4, Z/2Z) fixes (0, 0, 0, 0), it acts also on (Z/2Z)4 \{(0, 0, 0, 0)} and so we can identify it with a subgroup of O(AM(Z/2Z)4 ). This means that under the action of GL(4, Z/2Z) we have at most six orbits, associated to the cases 1;2b), 2a), 3a), 3b), 4a), 4b). We observe that the orbit of 2a) is the one of class 0 ∈ AM(Z/2Z)4 . We show now that all these orbits are disjoint, so we have exactly 6 (5 non trivial) orbits in ∨ M(Z/2Z) 4 /M(Z/2Z)4 . One can check by a direct computation that the classes of the cases 1) and 2b) coincide in the quotient. The classes in M(Z/2Z)4 with self intersection −2 are only ±M(a,b,c,d) , (a, b, c, d) ∈P (Z/2Z)4 \{(0, 0, 0, 0)}. Indeed each class in M(Z/2Z)4 is a linear combination D = (a,b,c,d)∈(Z/2Z)4 −{(0,0,0,0)} α(a,b,c,d) Ma,b,c,d P 2 with α(a,b,c,d) ∈ 21 Z. The condition −2 = D2 = −2 (a,b,c,d) α(a,b,c,d) implies that either there is one α(a,b,c,d) = ±1 and the others are zero, or there are four α(a,b,c,d) equal to ± 21 and the others are zero. Since there are no classes in M(Z/2Z)4 which are linear combination with rational coefficients of only four classes, we have D = ±M(a,b,c,d) for a certain (a, b, c, d) ∈ (Z/2Z)4 \{(0, 0, 0, 0)}. Since the isometries of M(Z/2Z)4 preserve the intersection product, they send the classes of the curves M(a,b,c,d) either to the class of a curve or to the opposite of the class of a curve. In particular, there are no isometries of M(Z/2Z)4 which identify classes associated to the six cases 1);2b), 2a), 3a), 3b), 4a), 4b), indeed in each class there is some linear combination with non integer coefficients of a different number of curves M(a,b,c,d) . Theorem 8.3. Let Y be a projective K3 surface such that there exists a K3 surface ^ 4 and let ρ(Y ) = X and a symplectic action of (Z/2Z)4 on X with Y = X/(Z/2Z) 16. Then N S(Y ) is generated by ZL ⊕ M(Z/2Z)4 with L2 = 2d > 0, and by a class ∨ 2 (L/2, v/2), 0 6= v/2 ∈ M(Z/2Z) ≡ −v 2 mod 8. Up to isometry 4 /M(Z/2Z)4 with L there are only the following possibilities:

30


¯ W where i) if d ≡ 1 mod 4, then v/2 = M W = {(1, 1, 0, 1), (1, 1, 1, 0), (1, 1, 1, 1), (1, 0, 0, 0), (0, 1, 0, 0)} (case 4b) of proof of Proposition 8.2); ¯ W where ii) if d ≡ 2 mod 4, then v/2 = M W = {(0, 0, 0, 1), (0, 0, 1, 0), (0, 0, 1, 1), (1, 0, 0, 0), (0, 1, 0, 0), (1, 1, 0, 0)} (case 4a) of proof of Proposition 8.2); if d ≡ 3 mod 4, then: either ¯ W where iii) v/2 = M W = {(0, 0, 0, 1), (0, 0, 1, 0), (0, 0, 1, 1)} (case 3b) proof of Proposition 8.2), or ¯ W where iv) v/2 = M W = (Z/2Z)4 − {(0, 0, 0, 0)} (case 1-2b)) proof of Proposition 8.2); ¯ W where v) if d ≡ 0 mod 4, then v/2 = M W = {(1, 1, 0, 0), (1, 1, 1, 0), (1, 1, 0, 1), (1, 1, 1, 1)} (case 3a) of proof of Proposition 8.2). Moreover for each d ∈ N there exists a K3 surface S such that N S(S) is an overlattice of index two of the lattice h2di ⊕ M(Z/2Z)4 . In cases i), ii), iii), v), TY ≃ U (2)⊕U (2)⊕h−2i⊕h−2di. In case iv) denote by q2 the discriminant form of U (2) then the of TY is (Z/2Z)5 ⊕ Z/2dZ discriminant group 0 1/2 . with discriminant form q2 ⊕ q2 ⊕ 1/2 (−d − 1)/2d Proof. In Proposition 8.1 we proved that the lattice N S(Y ) has to be an overlattice of index 2 of ZL ⊕ M(Z/2Z)4 . The unicity of the choice of v up to isometry depends on the description of the orbit of the group of the isometries of ∨ M(Z/2Z) 4 /M(Z/2Z)4 given in Proposition 8.2. By an explicit computation one can show that the discriminant group of the overlattices described in i), ii), iii), iv), v) is (Z/2Z)5 ⊕ (Z/2dZ) and the discriminant form in all the cases except iv) is q2 ⊕ q2 ⊕ 1/2 ⊕ 1/2d. In the case iv) the discriminant form is those described in the statement. In any case by [Ni2, Theorem 1.14.4 and Remark 1.14.5] the overlattices have a unique primitive embedding in the K3 lattice ΛK3 , hence by the surjectivity of the period map there exists a K3 surface S as in the statement of the theorem. Moreover by [Ni2, Theorem 1.13.2 and 1.14.2] the transcendental lattice is uniquely determined by signature and discriminant form. This concludes the proof. Remark 8.4. The Kummer surfaces appear as specializations of the surfaces Y as in Proposition 8.3 such that d ≡ 0 mod 2. Indeed, let us consider the surface Y such that d = 2d′ . The transcendental lattice of a generic Kummer of a (1, d′ )polarized Abelian surface is TKm(A) ≃ U (2) ⊕ U (2) ⊕ h−4d′ i, and it is clearly primitively embedded in TY ≃ U (2) ⊕ U (2) ⊕ h−2i ⊕ h−4d′ i.


31

8.1. Ampleness properties. As in Section 4, we can prove that certain divisors on Y are ample (or nef or nef and big) using the description of the Néron–Severi group of Y given in Theorem 8.3. The ample (or nef or nef and big) divisors define projective models, which can be described in the same way as in Section 5, where we described projective models of the Kummer surfaces. Proposition 8.5. With the notation of Theorem 8.3, the following properties for divisors on Y hold: • L is pseudo ample and it has no fixed components; • the divisor D := L − (M1 + . . . + Mr ), 1 ≤ r ≤ 15 is pseudo ample if d > r; ¯ := (L − M1 − . . . − Mr ) /2 ∈ N S(Y ) ⊗ Q; if D ¯ ∈ N S(Y ), then it is • let D pseudo ample if d > r. 8.2. K3 surfaces with 15 nodes. Here we show that a K3 surface with 15 nodes (resp. with 15 disjoint irreducible rational curves) is in fact the quotient (resp. the desigularization of the quotient) of a K3 surface by a symplectic action of (Z/2Z)4 . This is in a certain sense the generalization of a similar result for Kummer surface (cf. Section 2.2). Theorem 8.6. Let Y be a projective K3 surface with 15 disjoint smooth rational curves Mi , i = 1, . . . , 15 or equivalently a K3 surface admitting a singular model with 15 nodes. Then: 1) N S(Y ) contains the lattice M(Z/2Z)4 . 2) There exists a K3 surface X with a G = (Z/2Z)4 symplectic action, such that Y is the minimal resolution of the quotient X/G. Proof. 1) Let Q be the orthogonal complement in N S(Y ) to ⊕15 i=1 ZMi and R 15 be the lattice Q ⊕ ⊕i=1 ZMi . Observe that N S(Y ) is an overlattice of finite index of R and R∨ /R ∼ = Q∨ /Q ⊕ (Z/2Z)⊕15 so l(R) = l(Q) + 15. Let us denote by k the index of R in N S(Y ), thus l(N S(Y )) = l(Q) + 15 − 2k. On the other hand the rank of the transcendental lattice is 22 − rank(R) = 7 − rank(Q). Hence l(Q) + 15 − 2k ≤ 7 − rank(Q). Thus k ≥ (8 + l(Q) + rank(Q)) /2. We observe that k is the minimum number of divisible class we have to add to R in order to obtain N S(Y ). By definition the lattice Q is primitive in N S(Y ), thus the non trivial classes that we can add to R in order to obtain overlattices are either classes in (⊕i ZMi )∨ /(⊕i ZMi ) or classes like v + v ′ , where v ′ ∈ Q∨ /Q and v ∈ (⊕i ZMi )∨ /(⊕i ZMi ) is non trivial. By construction the independent classes of the second type are at most l(Q) and thus there are at least ((8 + l(Q) + rank(Q)) /2)− l(Q) = (8 + rank(Q) − l(Q)) /2 classes which are in (⊕i ZMi )∨ /(⊕i ZMi ). We recall that rank(Q) − l(Q) ≥ 0 and hence there are at least 4 classes which are rational linear combinations of the curves Mi . By [Ni1, Lemma 3] such a class in N S(Y ) can only contain 16 or 8 classes. Since 16 is not possible in this case, all these classes contain eight (−2)-curves. Let uj , j = 1, 2, 3, 4 be 4 independent classes in ∨ (⊕i ZMi ) / (⊕ZMi ) such that the uj are contained in N S(Y ). For each j 6= h, j, h = 1, 2, 3, 4, there are exactly 4 rational curves which are summands of both ui and uj , otherwise the sum ui + uj ∈ N S(Y ) contains half the sum of k ′ disjoint rational curves for k ′ 6= 8, which is absurd. It is now a trivial computation to show that there are at most 4 independent classes (and thus exactly 4) as required ∨ in (⊕i ZMi ) / (⊕ZMi ) and that for each choice of these 4 classes ui , the lattice

32


obtained adding the classes ui , i = 1, 2, 3, 4 to ⊕i ZMi is exactly M(Z/2Z)4 : indeed P8 without loss of generality the first class can be chosen to be u1 = i=1 (Mi /2), thus P4 P12 the second class can be chosen to be u2 = i=1 (Mi /2) + j=9 (Mj /2). The third class has 4 curves in common with u1 and with u2 and thus can be chosen to be u3 = (M1 + M2 + M5 + M6 + M9 + M10 + M13 + M14 )/2. Similarly, one determines the class u4 = (M1 + M3 + M5 + M7 + M9 + M11 + M13 + M15 )/2. 2) We consider the double cover π1 : Z1 −→ Y ramified on 2u1 . Since 2u1 contains 8 disjoint rational curves, Z1 is smooth. Moreover the pullback Ei of the curves Mi , i = 1, . . . , 8 have self intersection −1, hence these can be contracted to smooth points on a variety Y1 , and the covering involution that determines π1 descends to a symplectic involution ι1 on Y1 with 8 isolated fixed points (cf. [Mo, §3]). The divisors 2ui , i = 2, 3, 4 contain each 4 curves which are also in the support of 2u1 . We study the pull back of 2u2 , for the other classes the study is similar. We have 2π1∗ (u2 ) = π1∗ (2u2 ) = 2(E1 +. . .+E4 )+M51 +M52 +M61 +M62 +M71 +M72 +M81 +M82 , where π1 (Mji ) = Mj for i = 1, 2 and j = 5, 6, 7, 8. Hence the divisor M51 + M52 + M61 + M62 + M71 + M72 + M81 + M82 is divisible by 2 in N S(Z1 ) and so its image is divisible by 2 on N S(Y1 ). Doing the same construction as before, using this class we get a K3 surface Y2 with an action by a symplectic involution ι2 . Observe that ι1 preserves the divisor M51 + M52 + M61 + M62 + M71 + M72 + M81 + M82 and so ι1 and ι2 commute on N S(Y2 ). Considering now the pull-back of 2u3 and 2u4 on Y2 one can repeat the construction arriving at a K3 surface X := Y4 with an action by (Z/2Z)4 and such that the quotient is Y . We observe that the smooth model of a K3 surface admitting a singular model with 15 nodes contains 15 disjoint rational curves and we proved that such a K3 surface is a (Z/2Z)4 quotient of a K3 surface. Remark 8.7. Assume now that a K3 surface S either has a lattice isometric to M(Z/2Z)4 primitively embedded in the Néron–Severi group or its Néron–Severi group is an overlattice of Q ⊕ h−2i15 for a certain lattice Q. Then the Theorems 8.3, 8.6 do not imply that S is a (Z/2Z)4 quotient of a K3 surface. Indeed in the proof of Theorem 8.6, part 2), we used that the lattice h−2i15 (contained with index 24 in M(Z/2Z)4 ) is generated by irreducible rational curves. In other words the description of the Néron–Severi group from a lattice theoretic point of view is not enough to obtain our geometric characterization. Thus we can not conclude that the family of the K3 surfaces which are (Z/2Z)4 quotients of K3 surfaces coincides with the family of the K3 surfaces polarized with certain lattices. Remark 8.8. In the proof of Theorem 8.6 we proved that if a K3 surface contains 15 disjoint rational curves Mi , then there are 15 subsets Si , i = 1, . . . , 15 of 8 of these curves which form an even set. Similarly if a K3 surface has 15 nodes there are 15 subsets of 8 of these nodes which form an even set. In [Ba2] and [GSa1] some geometric properties of the even set of curves and nodes on K3 surfaces are described. For example if a quartic in P3 contains 8 nodes which form an even set, then the eight nodes are contained in an elliptic curve and there are three quadrics in P3 containing these nodes. Hence if a quartic in P3 has 15 nodes, each even set Si has the previous properties. Corollary 8.9. Let Y be a projective K3 surface with 14 disjoint smooth rational curves Mi , i = 1, . . . , 14. Then: 1) N S(Y ) contains the lattice M(Z/2Z)3 which is the minimal primitive sublattice


33

of the K3 lattice ΛK3 that contains the 14 rational curves. 2) There exists a K3 surface with a (Z/2Z)3 symplectic action, such that Y is the minimal resolution of the quotient of X by this group. Proof. The lattice M(Z/2Z)3 is described in [Ni3, Section 7]. The proof of 1) and 2) is essentially the same as the proof of 1) and 2) of Theorem 8.6. Remark 8.10. The result analogous to the one of Proposition 8.6 and Corollary 8.9 does not hold considering 8 (resp. 12) disjoint rational curves, i.e. considering the group Z/2Z (resp. (Z/2Z)2 ): 1) If a K3 surface is the minimal resolution of the quotient of a K3 surface by the group Z/2Z, then it admits a set of 8 disjoint rational curves but if a K3 surface admits a set of 8 disjoint rational curves, then it is not necessarily the quotient of a K3 surface by the group Z/2Z acting symplectically. An example is given by the K3 surface with an elliptic fibration with 8 fibers of type I2 and trivial Mordell– Weil group (cf. [Shim, Table 1, Case 99]): it contains 8 disjoint rational curves (a component for each reducible fibers), which are not an even set (otherwise the fibration admits a 2-torsion section). 2) If a K3 surface is the minimal resolution of the quotient of a K3 surface by the group (Z/2Z)2 , then it admits a set of 12 disjoint rational curves but if a K3 surface admits a set of 12 disjoint rational curves, then it is not necessarily the quotient of a K3 surface by a group (Z/2Z)2 acting symplectically. Anyway it is surely the quotient of a K3 surface by Z/2Z (the proof is again similar to the one of Theorem 8.6). An example is given by the elliptic K3 surface with singular fibers 2I0∗ + 4I2 which is the number 466 in Shimada’s list, [Shim]. The components of multiplicity 1 of the fibers of type I0∗ and a component for each fiber of type I2 are 12 disjoint rational curves. There is exactly one set of 8 of these curves which is a 2-divisible class (the sum of the components of the I0∗ fibers of multiplicity one). By using the Shioda-Tate formula one can easily show that there are no more divisible classes and hence the surface can not be the quotient of a K3 surface by (Z/2Z)2 . 9. The maps π∗ and π ∗ In the previous two sections we described the family of the K3 surfaces X admitting a symplectic action of (Z/2Z)4 and the family of the K3 surfaces Y obtained as desingularizations of the quotients of K3 surfaces by the group (Z/2Z)4 . Here we explicitly describe the relation among these two families. More precisely in Section e → Y , which of course induces the maps 6, we described the quotient map π : X π∗ and π ∗ among the cohomology groups of the surfaces: here we describe these maps (similar results can be found in [vGS] if the map π is the quotient map by a symplectic involution). With the notation of diagram (3) we have: e Z) −→ H 2 (Y, Z) is induced by the map Proposition 9.1. The map π∗ : H 2 (X, ⊕15 π∗ :< −2 >⊕16 ⊕U (2)⊕3 ⊕ < −1 >⊕8 π∗ : (k1 , . . . , k16 , u, {n1j }1≤j≤8 , . . . , {n15j }1≤j≤8 )

−→ 7→

< −2 > ⊕U (32)⊕3 ⊕ < −2 >⊕15 (k, u, m1 , . . . , m15 )

where π∗ (ki ) = k, for all i = 1, . . . , 16; π∗ (nij ) = mi for all j = 1, . . . , 8, i = 1, . . . , 15.

34


e Z) is induced by the map The map π ∗ : H 2 (Y, Z) −→ H 2 (X, π ∗ :< −2 > ⊕U (32)⊕3 ⊕ < −2 >⊕15 π ∗ : (k, u, m1 , . . . , m15 )

֒→ 7→

⊕15 < −2 >⊕16 ⊕U (2)⊕3 ⊕ < −1 >⊕8 P8 P8 (k1 = k, . . . , k16 = k, 16u, j=1 2n1j , . . . , j=1 2n15j )

Proof. By [Ni3, Theorem 4.7] the action of G on ΛK3 does not depend on the K3 surface we have chosen, hence we can consider X = Km(A) and G realized as in Section 3 (i.e. it is induced on Km(A) by the translation by the 2-torsion points of the Abelian surface A). 1. π∗ . We have seen that G leaves U (2)⊕3 invariant and in fact H 2 (X, Z)G ⊃ U (2)⊕3 , however the map π∗ multiply the intersection form by 16. In fact for x1 , x2 ∈ U (2)⊕3 we have: π ∗ π∗ (x1 ) = 16x1 so using the projection formula (π∗ x1 , π∗ x2 )Y = (π ∗ π∗ x1 , x2 )X˜ = 16(x1 , x2 ).

Since by taking X = Km(A) the classes in < −2 >⊕16 correspond to classes permuted by G their image by π∗ is a single (−2)-class in H 2 (Y, Z). Finally, the 120 (−1)-classes which are the blow up of the points with a non trivial stabilizer on X are divided in orbits of length eight and mapped to the same curve mi on Y . By using the projection formula and the fact that the stabilizer group of a curve nij has order 2, we have (mi , mi )Y = (π∗ (nij ), π∗ (nij ))Y = (π ∗ π∗ (nij ), nij )X˜ = (2(ni1 +. . .+ni8 ), nij )X˜ = −2. 2. π ∗ . Let x ∈ U (32)⊕3 and y ∈ U (2)⊕3 then (π ∗ x, y)X˜ = (x, π∗ y)Y = (x, y)Y = 16(x, y)X˜ so π ∗ (x) = 16x. Then we have π ∗ (u) = 16u since u is not a class in the branch locus. Finally (π ∗ (mi ), nhj )X˜ = (mi , π∗ (nhj ))Y = (mi , mh )Y = −2δih and (π ∗ (mi ), k)X˜ = (π ∗ (mi ), u)X˜ = 0 for u ∈ U (32)⊕3 . Hence π ∗ (mi ) is given as in the statement. Remark 9.2. The lattice R :=< −2 >⊕16 ⊕U (32)⊕3 (which is an overlattice of index 25 of K ⊕ U (32)⊕3 ) has index 223 in ΛK3 . Here we want to consider the divisible classes that we have to add to < −2 >⊕16 ⊕U (32)⊕3 to obtain the lattice ΛK3 . Consider the Z basis {ωij }i6=j of U (2)3 in H 2 (Km(A), Z). Recall that ·2

we have an exact sequance 0 → A[2] → A → A → 0, which corresponds to the multiplication by 2 on each real coordinates of A. Thus, the copy of U (32)⊕3 ⊂ H 2 (Km(A/A[2]), Z) is generated by 4ωij . Hence let ei , fi , i = 1, 2, 3 be the standard basis of each copy of U (32), then the elements: ei /4, fi /4 are contained in H (Y, Z). Adding these classes to R we find < −2 >⊕16 ⊕U (2)3 as overlattice of index 212 of R. In Remark 2.8 the construction of the even unimodular overlattice ΛK3 of h−2i⊕16 ⊕ U (2)⊕3 is described (we observe that the index is 211 ). In conclusion we can construct explicitly the overlattice ΛK3 of R and extend the maps, π∗ , π ∗ to this lattice. 2


35

10. Some explicit examples In this Section we provide geometrical examples of K3 surfaces X with Picard number 16 admitting a symplectic action of G = (Z/2Z)4 and of the quotient X/G, whose desingularization is Y . We follow the notation of diagram (3) and we denote by L the polarization on X orthogonal to the lattice Ω(Z/2Z)4 and by M the polarization on Y orthogonal to the lattice M(Z/2Z)4 . 10.1. The polarization L2 = 4, M 2 = L2 . We consider the projective space P3 and the group of transformations generated by: (x0 (x0 (x0 (x0

: x1 : x1 : x1 : x1

: x2 : x2 : x2 : x2

: x3 ) 7→ (x0 : x3 ) 7→ (x0 : x3 ) 7→ (x1 : x3 ) 7→ (x3

: −x1 : x2 : −x3 ) : −x1 : −x2 : x3 ) : x0 : x3 : x2 ) : x2 : x1 : x0 )

these transformations generate a group isomorphic to G = (Z/2Z)4 . The invariant polynomials are p0 p1 p2 p3 p4

= x40 + x41 + x42 + x43 = x20 x21 + x22 x23 = x20 x22 + x21 x23 = x20 x23 + x21 x22 = x0 x1 x2 x3

Hence the generic G-invariant quartic K3 surface is a linear combination: a0 (x40 +x41 +x42 +x43 )+a1 (x20 x21 +x22 x23 )+a2 (x20 x22 +x21 x23 )+a3 (x20 x23 +x21 x22 )+a4 x0 x1 x2 x3 = 0. Since the only automorphism commuting with all the elements of the group G is the identity, the number of parameters in the equation is 4, which is also the dimension of the moduli space of the K3 surfaces with symplectic automorphism group G and polarization L with L2 = 4. We study now the quotient surface. Observe that the quotient of P3 by G is the Igusa quartic (cf. [Hun, Section 3.3]), which is an order four relation between the pi ’s, this is: I4 :

16p44 + p20 p24 + p21 p22 + p21 p23 + p22 p23 − 4(p21 + p22 + p23 )p24 − p0 p1 p2 p3 = 0

Hence the quotient is a quartic K3 surface which is a section of the Igusa quartic by the hyperplane: a0 p0 + a1 p1 + a2 p2 + a3 p3 + a4 p4 = 0. The quartics in P3 admitting (Z/2Z)4 as symplectic group of automorphisms are described in a very detailed way in [E] (cf. also Remark 7.12). We observe that the subfamily with a0 = 0 is also a subfamily of the family of quartics considered by Keum in [Ke1, Example 3.3]. On this subfamily it is easy to identify an Enriques involution: this is the standard Cremona transformation (x0 : x1 : x2 : x3 ) −→ (1/x0 : 1/x1 : 1/x2 : 1/x3 ).

36


10.2. The polarization L2 = 8, M 2 = L2 /4 = 2. Let X be a K3 surface with a symplectic action of G and L2 = 8. There are two connected components of the moduli space of K3 surfaces with these properties (cf. Theorem 7.1 and Corollary 7.11). One of them is realized as follows. Let us consider the complete intersection of three quadrics in P5 :  P5  Pi=0 ai x2i = 0 5 bi x2i = 0  Pi=0 5 2 i=0 ci xi = 0.

with complex parameters ai , bi , ci , i = 0, . . . , 5. The group G is realized as the transformations of P5 changing an even number of signs in the coordinates. To compute the dimension of the moduli space of these K3 surfaces we must choose three independent quadrics in a six-dimensional space. Hence we must compute the dimension of the Grassmannian of the subspaces of dimension three in a space of dimension six. This is 3(6 − 3) = 9. Now the automorphisms of P5 commuting with the automorphisms generating G are the diagonal 6 × 6-matrices, hence we find the dimension 9 − (6 − 1) = 4 as expected. To determine the quotient, one sees that the invariant polynomials under the action of G are exactly the polynomials z02 , z12 , z22 , z32 , z42 , z52 and the product z0 z1 z2 z3 z4 z5 . Denote them by y0 , . . . , y5 , t then there is a relation t2 =

5 Y

yi ,

i=0

and so we obtain a K3 surface which is the double cover of the plane given by the intersection of the planes of P5 :  P5  Pi=0 ai yi = 0 5 bi y i = 0  Pi=0 5 i=0 ci yi = 0.

The branch locus are six lines meeting at 15 points, whose preimages under the double cover are the 15 nodes of the K3 surface. We get a special subfamily of K3 surfaces considering as in Section 5.1 a curve Γ of genus 2 with equation: y2 =

5 Y

(x − si )

i=0

with si ∈ C, si 6= sj for i 6= j. This determines a family of Kummer surfaces with (Z/2Z)4 action and equations in P5 :  2  z0 + z12 + z22 + z32 + z42 + z52 = 0 s0 z 2 + s1 z12 + s2 z22 + s3 z32 + s4 z42 + s5 z52 = 0  2 02 s0 z0 + s21 z12 + s22 z22 + s23 z32 + s24 z42 + s25 z52 = 0. Q The quotient surface also specializes to the double cover t2 = i yi of the plane obtained as the intersection of the planes of P5 :   y0 + y1 + y2 + y3 + y4 + y5 = 0 s 0 y0 + s 1 y1 + s 2 y2 + s 3 y3 + s 4 y4 + s 5 y5 = 0  2 s0 y0 + s21 y1 + s22 y2 + s23 y3 + s24 y4 + s25 y5 = 0.


37

As before the branch locus are 6 lines meeting at 15 points, but in this case there is a conic tangent to the 6 lines.

References [AN]

V.V. Alexeev, V.V. Nikulin, Del Pezzo and K3 surfaces. MSJ Memoirs, Vol. 15, Mathematical Society of Japan, Tokyo, 2006. [Ba1] W. Barth, Abelian surface with a (1, 2)-Polarization, Adv. Stud. Pure Math. 10, (1987) Algebraic geometry, Sendai, (1985), 41–84. [Ba2] W.Barth, Even sets of eight rational curves on a K3 surface. In: Complex Geometry (G¨ ottingen 2000), 1–25, Springer 2002. [BG] G. Bini, A. Garbagnati, Quotients of the Dwork pencil, arXiv:1207.7175. [BHPV] W. Barth, K. Hulek, C. Peters, A. van de Ven, Compact complex surfaces, Second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Band 4. Springer Berlin, 2004. [BS] S. Boissi` ere, A. Sarti, On the Neron–Severi group of surfaces with many lines, Proceedings of the American Mathematical Society, 136 (2008), 3861–3867. [BL] Ch. Birkenhake, H. Lange, Complex Abelian Varieties, Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 302. Springer-Verlag, Berlin, 2004. [E] D. Eklund, Curves of Heisenberg invariant quartic surfaces in projective 3-spaces, arXiv:1010.4058v2. [GaLo] F. Galluzzi, G. Lombardo, Correspondences between K3 surfaces (with an appendix by Igor Dolgachev) Michigan Math. J. 52 (2004), 267–277. [G1] A. Garbagnati, Symplectic automorphisms on K3 surfaces, PhD thesis (2008), University of Milan (Italy), available at https://sites.google.com/site/alicegarbagnati/phd-thesis. [G2] A. Garbagnati, Symplectic automorphisms on Kummer surfaces, Geom. Dedicata 145 (2010), 219–232. [G3] A. Garbagnati, Elliptic K3 Surfaces with abelian and dihedral groups of symplectic automorphisms, Comm. in Algebra 41 (2013), 583–616. [GH] P. Griffiths, J. Harris Principles of algebraic geometry. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978. [GSa1] A. Garbagnati, A. Sarti, Projective models of K3 surfaces with an even set, Adv. in Geom. 8 (2008), 413–440 [GSa2] A. Garbagnati, A. Sarti, Elliptic fibrations and symplectic automorphisms on K3 surfaces, Comm. in Algebra 37 (2009), 3601–3631. [vGS] B. van Geemen, A. Sarti, Nikulin involutions on K3 surfaces, Math. Z. 255 (2007), 731– 753. [GrLa] R. L. Griess Jr., C.H. Lam, EE8 -lattices and dihedral groups Pure Appl. Math. Q. 7 (2011), Special Issue: In honor of Jacques Tits, 621–743. [Ha] K. Hashimoto, Finite symplectic actions on the K3 lattice, Nagoya Math. J. 206 (2012) 99–153. [Ho] E. Horikawa, On the periods of Enriques surfaces. I, II, Math. Annalen 234 (1978), 73–88; 235 (1978), 217–246. [Hud] R. W. H. T. Hudson, Kummer’s quartic surfaces, with a foreword by W. Barth. Revised reprint of the 1905 original. Cambridge University Press, Cambridge, (1990). [Hun] B. Hunt, The Geometry of some special Arithmetic Quotients, Lecture Notes in Mathematics 1637, Springer-Verlag Berlin-Heidelberg (1996). [HS] K. Hulek, M. Sch¨ utt, Enriques Surfaces and jacobian elliptic K3 surfaces, Math. Z. 268 (2011), 1025–1056. [I] H. Inose, On certain Kummer surfaces which can be realized as non-singular quartic surfaces in P3 , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23, 23, (1976), 545–560. [Ke1] J. Keum, Automorphisms of Jacobian Kummer surfaces, Compositio Math. 107 (1997), 269–288. [Ke2] J. Keum, Every algebraic Kummer surfaces is the K3-cover of an Enriques surface, Nagoya Math. J. 118 (1990), 99–110. [KK] J. Keum, S. Kondo The automorphism groups of Kummer surfaces associated to the product of two elliptic curves, Transactions of the AMS 353, No. 4 (2001), 1469–1487.

38


[Koi]

K. Koike, Elliptic K3 surfaces admitting a Shioda–Inose structure, Comment. Math. Univ. St. Pauli61 (2012) 77–86. [Kon] S. Kondo,, The automorphism group of a generic Jacobian Kummer surface J. Alg. Geom. 7 (1998), 589–609. [Kum1] A. Kumar, K3 Surfaces Associated with Curves of Genus Two, Int. Math. Res. Not. (2008), 1073–7928. [Kum2] A. Kumar, Elliptic fibrations on a generic Jacobian Kummer surface, arXiv:1105.1715, accepted for publication in J. Algebraic. Geom. [KSTT] K. Koike, H. Shiga, N. Takayama, T. Tsutsui, Study on the family of K3 surfaces induced from the lattice (D4 )3 ⊕ h−2i ⊕ h2i, Internat. J. Math. 12 (2001), 1049-1085. [Me] A. Mehran, Kummer surfaces associated to a (1, 2)-polarized abelian surfaces, Nagoya Math. J. 202 (2011), 127–143. [Mo] D. Morrison, On K3 surfaces with large Picard number, Invent. Math. 75 (1984), 105–121. [Mu] S. Mukai, Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94 (1988), 183–221. [Na] I. Naruki, On metamorphosis of Kummer surfaces, Hokkaido Math. J., No. 20, (1991), 407–415. [Ni1] V. V. Nikulin, Kummer surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 278–293. English translation: Math. USSR. Izv, 9 (1975), 261–275. [Ni2] V.V. Nikulin: Integral Symmetric Bilinear Forms and Some of their Applications, Math. USSR Izvestija Vol. 14 (1980), No. 1, 103–167. [Ni3] V.V. Nikulin: Finite groups of automorphisms of K¨ ahlerian surfaces of type K3, Trudy Mosk. Mat. Ob. 38, 75–137 (1979); Trans. Moscow Math. Soc. 38, 71–135 (1980). [O] K. Oguiso, On Jacobian fibrations on the Kummer surfaces of the product of nonisogenous elliptic curve, J. Math. Soc. Japan 41 (1989), 651–680. ¨ [OS] H. Onisper, S. Sert¨ oz, Generalized Shioda-Inose structures on K3 surfaces, Manuscripta Math. 98 (1999), 491–495. [PS] I. Piatetski-Shapiro, I.R. Shafarevich, Torelli’s theorem for algebraic surfaces of type K3, Izv. Akad. Nauk. SSSR Ser. Mat. 35 (1971), 530–572. English translation: Math. USSR, Izv. 5 (1971), 547–587. [SD] B. Saint-Donat, Projective models of K3 surfaces, Amer. J. of Math. 96 (1974) 602–639. [Shim] I. Shimada, On Elliptic K3 surfaces, Michigan Math. J. Volume 47, Issue 3 (2000), 423– 446, arXiv:math/0505140v3. [Sh1] T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59. [Sh2] T. Shioda, Some results on unirationality of algebraic surfaces, Math. Ann. 230 (1977), 153–168. [Sc] M. Sch¨ utt, Sandwich theorems for Shioda-Inose structures, Izvestiya Mat. 77 (2013) 211– 222. [X] G. Xiao, Galois covers between K3 surfaces, Annales de l’Institut Fourier 46 (1996), 73–88. ` di Milano, via Saldini 50, Alice Garbagnati, Dipartimento di Matematica, Universita I-20133 Milano, Italia E-mail address: [email protected] ´ de Poitiers, Laboratoire de Mathe ´matiques et AppliAlessandra Sarti,Universite ´ le ´port 2 Boulevard Marie et Pierre Curie BP 30179, cations, UMR 6086 du CNRS, Te 86962 Futuroscope Chasseneuil Cedex, France E-mail address: [email protected] URL: http://www-math.sp2mi.univ-poitiers.fr/~sarti/